Exploring the Boundaries of Modern Quantum Annealers with RNA Structure Prediction

Abstract

Quadratic unconstrained binary optimization (QUBO) has gained popularity over the past ten years both for the wide range of problems that can be expressed in this form and for its compatibility with many quantum computing architectures. More recently, quantum annealers have advanced to the point where the scaling of qubits makes real-world applications increasingly feasible; however, current architectures remain restricted to solving quadratic unconstrained binary optimization problems, which can represent at most two-body interactions. Here we systematically examine the shortcomings of the QUBO framework and modern quantum annealers through the lens of the high-dimensional, exponentially scaling problem of RNA structure prediction, and find results that are generalizable to QUBO formalisms writ large. We find that using the QUBO framework to predict RNA structures results in poor accuracy, which becomes worse as system size scales. We determine that the fundamental constraint of two-body interactions prevents emergent correlations in the system, which are crucial to accurate structure prediction. Furthermore, we highlight that explicit, higher-order, empirical terms that are needed to improve structure prediction are incompatible with a QUBO framework. Finally, we show that our energy landscape has many deep local minima, which lead to local minima trapping and prevent permutation invariance. The relevance of these findings to other applications of QUBO, and therefore to modern quantum annealers more broadly, are discussed, with our work providing the first explicit explanation of why these formalisms have failed and will continue to fail in high-dimensional systems with similar constraints.

Introduction

Quantum computing has been theorized to have significant advantages over classical computing in solving complex combinatorial optimization problems with intractably large phase spaces. Many combinatorial problems are NP-hard and, as such, exhibit exponential growth in complexity as a function of input size. These problems are therefore often impractical to solve on classical computers. However, quantum computers have been shown to outperform their classical counterparts in several optimization tasks − due to their ability to reduce time complexity and transform these classically intractable problems into tractable ones as input size grows. Although there are several types of quantum computing architectures, quantum annealing systems already feature thousands of qubits and are rapidly expanding, unlike most other platforms, which typically have only tens or hundreds of qubits. Recently, Kim et al. benchmarked the performances of D-Wave’s quantum annealer and classical solvers on a variety of optimization tasks and demonstrated a dramatic reduction in problem-solving time when using quantum hardware. Furthermore, in March of 2025, D-Wave Inc. demonstrated quantum supremacy by performing a spin-glass calculation using their quantum annealer that is beyond the practical capabilities of any classical computer. The recent successes of quantum annealing hardware, it both hardware scaling and application to modern problems, make it an attractive system for testing the limits and applicability of current quantum computing systems to a variety of large-scale optimization problems.

Current quantum annealing architectures, such as those implemented by D-Wave, require that problems be encoded in the Quadratic Unconstrained Binary Optimization (QUBO) formalism. QUBO is a mathematical framework used to express NP-hard combinatorial optimization problems with binary variables into a quadratic objective function. It has the powerful ability to translate diverse optimization problems into a single and simple framework. Therefore, it provides an attractive method to solve NP-hard problems in many applications with intractably large phase spaces. − In this study, we investigate the implications of applying this method to a biologically relevant problem, RNA secondary structure prediction.

RNA folding is a process driven by the formation of hydrogen bonds between base pairs in a single-stranded RNA sequence. There are four bases (A, U, G, C), and each base can interact with only one other base in the sequence through either Watson–Crick (G-C, A-U) and Wobble (G-U) pairing. Consecutive base pairs form a stem, and other secondary structure elements, such as loops, emerge from unpaired regions between or within stems. RNA structure is thus dictated by the sequence of bases and set of possible base pairs, which determine a unique combination of stems and emergent three-dimensional structure. RNA structure prediction can therefore be framed as a combinatorial optimization problem of selecting the optimal set of stems. While this framework initially appears simple to solve due to the simple RNA folding rules, critically, the phase space grows exponentially as you increase the size of the input (in this case, the RNA sequence length). For a given sequence, there are 2 ^N – C possible structures where N is the number of possible stems and C is the number of stem pairs which are excluded because they overlap (i.e., would share bases if both were present in structure). The Pfizer-BioNTech COVID-19 mRNA vaccine consists of about 4000 bases and approximately 5 × 10⁵ possible stems. It is impossible to exhaustively search such large phase spaces using classical computers. Therefore, RNA folding is an ideal candidate for testing the utility of the QUBO framework and modern quantum annealers in large scale optimization problems.

RNA folding has been framed as a QUBO problem in prior works, and structures of short RNA sequences, 30–75 bases in length, have been accurately predicted using quantum computing hardware. − While this method has proven successful for short RNA sequences, it must be validated on longer sequences that are more relevant in biological processes and the design of mRNA vaccines. In this work, we therefore employ a variation of the Hamiltonian from Fox et al. to extend the use of the QUBO formalism to predict the structures of 2738 sequences of up to 300 bases in length from the archiveII database. This allows us to evaluate the effectiveness of using QUBO frameworks, and, by extension, modern quantum computing architectures, for a high-dimensional optimization problem. We demonstrate that embedding RNA secondary structure prediction into a QUBO framework imposes fundamental limitations on model performance. Although inspired by successful applications of QUBO to small sequences, our results on larger sequences reveal that predictive accuracy rapidly degrades as sequence length increases. We show that this failure is driven by a rugged energy landscape with deep, narrow local minima and high energy barriers that hinder optimization. Critically, the QUBO formalism’s restriction to one- and two-body terms prevents the inclusion of essential constraints required to reproduce the sparsity, heterogeneity, and higher-order structural features of real RNA. Finally, the model fails to generate the long-range correlations necessary for higher-order structure to emerge, further limiting its applicability to biologically relevant systems. Together, our results highlight that the QUBO framework, in its current form, is structurally incapable of accurately capturing RNA folding at biologically relevant scales and challenges the presumed utility of modern quantum annealers for problems with exponentially scaling phase spaces such as the optimization of complex molecular structure.

Methods

Hamiltonian Construction

The Hamiltonian is formulated as described in Fox et al. First, all possible stems for a given RNA sequence are enumerated by constructing an N × N matirx, where both dimensions index the same N nucleotide sequence. An off-diagonal entry (i, j) is set to 1 if nucleotide i can form a valid base pair with nucleotide j, and 0 otherwise. Stems are then identified by finding consecutive values of 1 along the axis that is perpendicular to the matrix diagonal, and the list of valid stems is compiled. The minimum length of a valid stem is treated as a hyperparameter. After all stems are enumerated, an M × M matrix is defined in which both axes index the same set of M stems. The diagonal entries encode the energetic contribution of each individual stem, while the off-diagonal entries encode the energetic contributions arising from pairs of stems i and j. The numerical values in the matrix are determined by the objective function, defined below. This matrix defines the free energy landscape and is an input to the quantum annealer.

Objective Function

This work follows the method proposed by Fox et al. and will be briefly described here. First, all possible stems, of minimum length three and resulting loops of minimum length three and involving only Watson–Crick (A-U, G-C) and Wobble base pairs (G-U), are enumerated. The Hamiltonian is designed to be solved as a Quadratic Unconstrained Binary Optimization model where the bits represent the presence of the stem in the structure. The Hamiltonian is then

$$H=\sum _i(c_L{(k_i-\mu )}^2-c_B{k}_i^2)q_i-\sum _{j,k,k>j}(c_B{k}_j{k}_k{\delta }_{\mathit{jk}}+{\delta }_{\mathit{jk}}^{\prime })q_j{q}_k$$ 1

where k _i is the length of stem i and q _i represent bits (0 or 1) indicating if stem i is in the structure. The first term, c _L (k _i – μ)², penalizes stems which have lengths that deviate from μ. The second term, −c _B k _i , promotes the formation of sequential base pairs (i.e., longer stems). The third, c _B k _i k _j δ _jk , represents a pseudoknot penalty where δ _jk is 1 if stems j and k do not form a pseudoknot and 0.5 if stems j and k do form a pseudoknot. The final term, δ _jk , assigns an “infinite” energy if stems j and k overlap or share the same bases.

In this work, an additional term is added to limit the number of stems which can be selected for each structure as an equality constraint in the form of

$$\begin{array}{ll}{H}_P=A{(\sum _i{q}_i-\mathit{fN})}^2& (2)\\ =-2AfN\sum _i{q}_i+A\sum _{j,k,k>j}{q}_j{q}_k+A{(\mathit{fN})}^2& (3)\end{array}$$ 3

where A is a constant, f is the fraction of desired bits to be assigned a value of 1 after optimization, and N is the sequence length in bases. Adding eq 2, excluding the final constant, to eq leads to the equation

$$H=\sum _i(c_L{(k_i-\mu )}^2-c_B{k}_i^2-2\mathit{AfN})q_i-\sum _{j,k,k>j}(c_B{k}_j{k}_k{\delta }_{\mathit{jk}}+{\delta }_{\mathit{jk}}^{\prime }+A)q_j{q}_k$$ 4

The optimal RNA structure is then determined by identifying combination of stems which minimize eq . To do this, we use D-wave’s simulated annealing sampler from the dwave-samplers 1.2.0 python package. Each structure is determined after 50,000 iterations unless stated otherwise. This method is used to predict the structures of sequences present in the archiveII data set with 300 or fewer bases (2783 of the 3975 database structures). This work investigates larger sequence lengths than explored by previous works − by a factor of 3–4. Structure predictions are performed with four sets of parameters which are listed in Table S1.

Evaluating Predicted Structures

Each predicted structure is then directly compared to the experimental structures in the archiveII database. Quality of predicted structures are judged using the confusion matrix and other derived metrics such as sensitivity, specificity, and F1 score as described in Mathews and are shown in eqs –, respectively.

$$\mathrm{s}\mathrm{ensitivity}=\frac{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{s}}{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{s}+\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{negatives}}$$ 5

$$\mathrm{s}\mathrm{pecificity}=\frac{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{s}}{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{s}+\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{posit}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{s}}$$ 6

$$\mathrm{F1}=\frac{2\times \mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{s}}{2\times (\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{s}+\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{s}+\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{negatives})}$$ 7

A true positive is a correctly identified base pair, a true negative is a correctly identified unpaired base, a false positive is a base which is paired to a different base than observed in the experimental structure or an unpaired base which has a pair in the predicted structure, and a false negative is a base which is unpaired in the predicted structure but paired in the experimental structure. Note that, in this case, false positives can represent two possible scenarios so comparing to experimental structures is not strictly a binary assessment. This means the sum of true positives and false negatives will not be the same value for different predicted structures of the same sequence. However, the sum of all four values of the confusion matrix will be the same for each predicted structure of the same sequence.

Investigating Permutation Invariance and Local Minimum Trapping

A set of random sequences of different sequence lengths from the archiveII data set were selected. For each sequence, all of the stems would be generated as described above. Ten sequential structure predictions were made. In between each structure prediction iteration, the list of stems which were used to construct the Hamiltonian matrices would be permuted. F1 scores were computed for each unique pair of the resultant ten structures, rather than the experimental structure, to investigate diversity of predicted structures.

Exploring Effects of Stem Overlaps on Emergent Long-Range Correlations with Metropolis Monte Carlo Simulations

To investigate the direct effect of stem overlaps, or pairwise exclusions, in a sequence independent way, a system of all-to-all connected 50 bits are allowed to evolve according to eq using the Metropolis algorithm. The simulation is performed for 500,000 iterations at a temperature of 0.01, which is below the expected critical temperature for an analogous spin system. Each bit is assigned a random k _i value in the range [1, 5], to emulate different stem lengths, and the following hyperparameters are used: c _B = 1, c _L = 0, A = f = 0, and the pseudoknot factor is excluded. In each simulation, an increasing fraction of the total number of pairs of bits are disallowed, or pairwise excluded, from both having values of 1. This is meant to simulate the effect of the pairwise exclusion constraint (i.e., or the effect of RNA stem overlaps) on the ability for long-range correlations to emerge within the system. To quantify this, the order parameter (O) and average pair correlations (PC), are calculated by averaging the sum of the bit values and the product of the bit values, respectively, as in eqs and . every 100 iterations for the final 200,000 iterations of the simulations.

$$O=⟨q_i⟩=\frac{1}{N}\sum _i{q}_i$$ 8

$$⟨\mathrm{PC}⟩=\frac{1}{N}\sum _{i,j,j>i}⟨q_i{q}_j⟩$$ 9

Results

QUBO Model Fails to Reproduce Accurate RNA Structures

We developed a Hamiltonian modified from Fox et al. to explore the RNA-folding phase space using D-wave’s simulated annealing sampler. This allowed us to test the performance of RNA structure prediction using a QUBO formalism on sequences from the archiveII data set of up to 300 bases in length. Following Fox et al., we constructed a Hamiltonian with binary variables that represent the inclusion or exclusion of a possible stem. Our model is inherently quadratic and therefore can only capture properties of single stems and pairwise relationships between them. The combination of stems that minimizes this Hamiltonian represents a predicted RNA structure. The first one-body term in our model has a quadratic form that penalizes stems with lengths that deviate from μ. The second one-body term promotes the inclusion of base pairs in a stem and encourages the model to maximize total base pairing. The first two-body term promotes longer stem pairs and imposes a penalty for selecting two stems that form a pseudoknot (structural motifs that form when bases in a loop pair with bases outside that loop); however, it allows the model to include pseudoknots at a cost, rather than forbidding them outright. The final two-body term prohibits stem overlaps, which are unphysical since a nucleotide can typically only participate in one base pair. Our initial structure prediction results motivated the addition of a term that limits the number of stems in each structure.

We next empirically selected four sets of hyperparameters and designed additional constraints to explore our method’s structure prediction capabilities (Table S1). Each hyperparameter set was selected to target different configurational biases to evaluate how hyperparameter tuning and added constraints affect emergent structural properties and predictive performance. We adopted the hyperparameter set used in Fox et al. as set I, given its demonstrated success in accurately predicting RNA structures for short sequences of 30–75 nucleotides. This hyperparameter set promotes the formation of long stems. Set II allows for the selection of shorter stems, which is achieved by reducing the strength of the quadratic penalty term, normally used to constrain stem length, and by lowering the target stem length parameter, μ, to the average length of the stems in the experimental data set. Set III enforces a limit on the number of selected stems through a quadratic penalty on their total count. Finally, set IV imposes a 100-nucleotide span limit between paired bases, following Lange et al., who demonstrated that other RNA structure prediction algorithms improved when implementing a base-pair interaction range restriction of 100–150 bases. For more details on these selections of parameters and constraints, refer to the Supporting Text. Together, these four hyperparameter sets allow us to systematically investigate how structural biases and constraint strategies shape the energy landscape and influence our model’s ability to successfully predict RNA structures from their sequences.

The Hamiltonian is constructed classically by first enumerating all stems and subsequently generating an M × M stem matrix where the diagonal entries represent the one-body contributions for stem i and the off-diagonal terms represent pairwise contributions for stems i and j, as described in the Methods section. The Hamiltonian is then passed to the annealer as the input. We used D-wave’s simulated annealing sampler on 2738 sequences from the archiveII data set for each hyperparameters set and computed F1 scores (Figure ), the sensitivity, and the specificity of all our predicted structures (see Figure S2) to evaluate how closely they match biological reality. Sensitivity measures the fraction of base pairs that are correctly predicted, specificity measures the portion of true negatives identified, and F1 measures how many base pairs were correctly predicted and how well incorrect pairs were avoided. The average value of these metrics, computed from per-sequence values, are used to compare the predictive performance of each hyperparameter set. Parameter set I produced an average F1 score of 0.41, sensitivity of 0.52, and specificity of 0.44. Parameter Set II produces a lower F1 score of 0.19. While the set II sensitivity value is higher than for set I at 0.56, the specificity is 0.13, indicating a large prevalence of false positives in structures from set II. Parameter set III performs slightly better than set II with an F1 score of 0.24. Set IV has the worst predictive performance, with an F1 score of 0.16. Interestingly, set IV exhibits low specificity, like sets II and III, but has lower average sensitivity than all other parameter sets underscoring the importance of including long-range interactions in structure optimization. Overall, our F1 scores indicate poor predictive capabilities and are lower than those of various other machine learning or physics-based RNA structure prediction methods whose average F1 scores are 0.554–0.905 for the same RNA data set. We further explored two hyperparameter optimization strategies, but they failed to improve predictive performance (Supporting Text).

1.

Distributions of F1 scores and example experimental and predicted RNA structures. (A) F1 scores, calculated with respect to the experimental structures, of predicted structures for RNA sequences of 300 or less base pairs from the archiveII data set for four sets of hyperparameters (Table S1). Visualizations of experimental and predicted RNA structures for (B) 5s_Acidithiobacillus-thiooxidans-1, (C) 5s_Colwellia-psychrerythraea-1, and (D) srp_Leis.braz._AY722724. Visualizations were made with RNAcanvas.

Visual inspection of three representative sequences showed that our model produces highly inconsistent structures across different hyperparameter sets (Figure ). The first two rows of Figure B–C show two experimental structures for two RNA sequences that differ by only four of their 119 bases. While the resulting experimental structures are very similar, our model additionally fails to exhibit consistency within a single hyperparameter set. Finally, Figure D highlights that our model fails to correctly reproduce experimental structures that have very few stems.

Since our study includes sequences longer than those in prior studies, we also investigated predictive performance as a function of sequence length to understand how system scale affects predictive strength. Figure S3 shows that, with all four hyperparameter sets, F1 scores rapidly decrease as the sequence length increases. To understand the scaling properties of the problem, we characterized the phase space for all 3000 sequences in the archiveII data set (Figure S4A,B and E) which demonstrates the number of possible stems which could be chosen for an RNA structure increases exponentially and the distribution of possible stem lengths widens with increasing sequence length. Based on the resulting F1 scores, the QUBO-based model presented here cannot adequately handle the extreme scaling of the problem. Together, our quantitative and qualitative analyses show that the QUBO-based model fails to capture consistent and accurate RNA structures, and its predictive strength will dramatically decrease as the input sequence length scales to biologically relevant sequence lengths.

Deep Local Minima Dominate the Energy Landscape and Undermine the Utility of Structure Prediction Using QUBO

While our structure prediction results demonstrate that our QUBO model fails to accurately and consistently reproduce archiveII data set RNA structures from sequences of up to 300 bases, they do not fully explain why the model performs so poorlyeven with varied hyperparameters and biologically informed constraints. Therefore, we next examine properties of the energy landscape defined by the QUBO Hamiltonian to search for sources of error. The energy landscape can be traversed by adding or removing stems, and the resulting energy changes provide insight into the landscape’s overall shape. We unfolded predicted structures by removing stems in a random order and recorded the energy change after each step. This procedure was repeated five times, and stems were removed in different random orders to mimic the effects of different folding pathways. Figure A,B shows the results of this unfolding process for the 5s_Acheta-domesticus-1 sequence for each hyperparameter set. Figure A shows the average change in energy from adding subsequent stems is shown and Figure B shows the sum of those averages, showing the steep gradients of the free energy landscape near a local or global minimum. The error bars represent the spread of the change in energy as a result of sampling the five different unfolding pathways. Although the analysis was performed by unfolding, folding and unfolding are symmetric processes along the same pathway, so the changes in energy as a function of removing or adding stems are identical. We found that as more stems are added, the magnitude of the energy change generally increases or remains constant. Therefore, leaving a local minimum by removing stems becomes more difficult as sequences get longer and can accommodate more stems. These results reveal that our QUBO formalism defines an energy landscape with steep gradients and fluctuating topography, which makes the optimizer very susceptible to local minimum trapping and hinders its ability to find the structure associated with the global minimum. This issue is furthermore exacerbated as sequence lengths increase.

2.

Demonstrating properties of phase space for the 5s_Acheta-domesticus-1 sequence of the archiveII data set. (A) The change in energy (dE) vs the number of stems added to a structure for predicted structures derived from each hyperparameter set used in this work. Data points are averages and error bars are SEM of five unfolding pathways where stems were removed from the structure in five random orders. (B) Total energy (E) vs the number of stems calculated using the averages in (B). (C) A transition diagram between the predicted structure of 5s_Acheta-domesticus-1 with parameter set II shown in the bottom left. Bottom right shows the same structure but the bottom right stem has one fewer base pair (3 instead of 4). Visualizations of RNA structures were made with RNAcanvas.

Our embedding choice introduces a critical limitation that further complicates traversing the energy landscape: similar RNA configurations are separated by large energy barriers. Figure C illustrates this effect through two transition pathways between (i) the structure of 5s-Acheta-domesticus-1 predicted using hyperparameter set II and (iii) a nearly identical structure with one base pair removed. Because our model maps each qubit to a stem, navigating the energy landscape is limited to stem-level changes, and individual base pairs cannot be removed or added. Therefore, transitioning from (i) to (iii) requires passing through an intermediate structure (ii). In this pathway, the original stem is first removed (dE = 384) and then a new stem, which is one base pair shorter, is added (dE = −284). Although structures 1 and 3 differ by only 100 energy units, the intermediate step incurs a much larger energy cost (over three times greater) creating an artificial barrier that would not exist in a continuous or physics-based energy model. As structures grow in complexity and include more stems, these barriers scale disproportionately, forming increasingly deep energy wells surrounded by steep walls. This feature of the QUBO embedding exacerbates local trapping and makes it far more difficult to escape suboptimal configurations than would be expected in a physically realistic folding landscape.

We observe evidence of deep local minima in our model. To assess whether they affect structure prediction, we performed a test of permuting the order of candidate stems in the input. Mathematically, reindexing the stems does not alter the underlying free energy landscape. In practice, however, reordering the stems after enumeration changes the ordering of terms in the Hamiltonian matrix. Upon initialization, the optimizer selects an initial structure based on a random seed. For a fixed seed, the same indices are chosen, but if stems are reordered, those indices correspond to a different physical structure and therefore a different local neighborhood of the energy landscape. If the energy landscape is easily traversed, this reordering should have no effect in practice, leading to a permutation invariant model which can find the same minimum regardless of input indexing. To explore this permutation invariance, ten sequential structure predictions were performed for a given RNA sequence. Between each prediction, the order of the list of stems was shuffled and the matrices were reconstructed. The pseudorandom number generator was not reset between predictions. To demonstrate the variety of predicted structures, F1 scores between all unique pairs of the ten predicted structures were calculated, averaged, and the error was computed as the standard error of the mean (SEM). Note that no comparison was made to the experimental structure in this case because different predicted structures could potentially produce similar F1 scores when compared to the same reference structure. The goal is to capture the similarity, or lack thereof, of predicted structures for the same sequence given permuted inputs. This was done for a set of seven sequences varying in length (see Table S3) and the results are shown in Figure .

3.

Average F1 score, calculated by comparing sets of ten predicted structures with permuted inputs, for increasing sequence lengths and each set of hyperparameters. Sequences were chosen randomly from the archiveII data set and are listed in Table S3 and parameters are listed in Table S1. Note that F1 score here is not a comparison to experimental data but instead a comparison of the set of predicted structures produced by permuting the input matrix of the QUBO Hamiltonian. High average F1 score then indicates the predicted structures were similar while low average F1 score indicates the opposite. Error bars are calculated as SEM of the ten F1 scores. Low F1 scores are indicative of severe local minima trapping and that the output of the model is highly sensitive to the order of the inputs (and therefore the starting location in phase space) which is undesirable.

These results show that, for each set of hyperparameters, increasing sequence length results in lower average F1 scores between predicted structures. This indicates that permuting the inputs alone can lead to drastically different predicted structures. We also explored whether increasing the number of annealing iterations would alleviate these issues. Figure S7 shows how increasing the number of iterations in the annealing process effects the resultant energy and F1 scores for ten predicted structures for the longest sequence considered in this analysis (297 bases). Specifically, it shows that increasing the number of iterations does not increase the average F1 score significantly and that the SEM of the F1 score sometimes increases with iterations, pointing to a wider variety of structures. Additionally, ten sequential predictions were performed without shuffling the stem lists, and all ten structures were identical (F1 = 1.0) for two sets of predictions with different values for μ for each number of optimization iterations performed (Figure S7). This further demonstrates that our predictions are deterministic if we begin traversing phase space from the same location, and the effect observed in Figure is not the result of too few annealing iterations or the influence of the state of the pseudorandom number generator during optimization (as it was not reset between optimizations). The deep minima of the free energy landscape that lead to this phenomenon also clearly contribute to the large structural changes observed from the few point mutations between Figure B,C as point mutations are analogous to rearranging phase space in this model.

Missing Physics: QUBO’s Structural Limitations

While the previous sections demonstrated that our model performs poorly and is hindered by a rugged energy landscape with deep local minima and high energy barriers, these challenges alone do not fully account for its failure to predict accurate RNA structures. We next probe stem features in our predicted structures to highlight the ways in which our QUBO model fails to recapitulate distributions seen in experimental structures. First, our model typically overpredicts the number of stems (Figure A), base pairs (Figure B), and pseudoknots (Figure S5 and Table S2). In Figure A,B, each data point represents a predicted structure of one of 2783 RNA sequences from the archiveII data set, and the red line is a reference line with slope unity. A data point that lies on the red line indicates the number of stems or base pairs in the predicted structures equals those of experimental structures. Generally, our data points fall above the red line, indicating an overprediction of the number of stems in RNA structures (Figure A). Hyperparameter set I is an exception. It underpredicts the number of stems because it selects for longer stems, fewer of which can be accommodated within a structure, resulting in a lower total stem count. However, Figure B shows that the predicted structures from all hyperparameter sets tend to drastically overestimate the number of base pairs, sometimes by a factor of 2. Additionally, Table S2 and Figure S5 show that our predicted structures contain many more pseudoknots than are in experimental structures, and Figure B–D highlights a handful of examples of this.

4.

Comparison of stems and base pairs between experimental and predicted structures of sequences of 300 or fewer bases from the archiveII data set and predicted structures. Predicted vs experimental (A) number of stems and (B) number of base pairs for each set of hyperparameters (columns, Table S1). (C) Distributions of predicted and experimental stem length ranges, the difference between largest and smallest stem length in base pairs, for all sequences and hyperparameter sets considered. (D) Distributions of predicted and experimental stem lengths, in base pairs, for the srp_Xeno.laev._X01055 sequence of the archiveII data set.

We also wanted to compare the distributions of predicted stem lengths to those observed in the experimental structures. The results are shown in Figure S6 and demonstrate our model only ever predicts structures with narrow unimodal distributions of stem lengths centered close to the value of μ. Since these results consider all stems from all structures, some detail is lost. To obtain more information on a per-sequence basis, we also calculated the distribution of stem length ranges, smallest stem length subtracted from largest stem length in base pairs, to show how the distributions of stem lengths can be narrow and wide. The results are shown in Figure C and demonstrate that experimental structures have, on average, a larger differences between their largest and smallest stems than in any of our predicted structure sets. Larger stem length ranges in experimental structures can be attributed to system scaling. Figure S4B,C shows the largest observed stem length grows with sequence length while the average converges to a low value, indicating a widening distribution of stem lengths with increasing sequence length. Figure D shows the predicted and experimental stem length distributions for the srp_Xeno.laev._X01055 sequence, containing 297 bases, and shows that none of the predicted structures capture the multimodal distribution of stem lengths observed in the experimental structure. Together, this indicates a successful QUBO model has to be able to accommodate varying stem length distribution shapes (narrow and unimodal, wide and unimodal, or multimodal) on a per-sequence basis. The distributions from our predicted structures show our model generally produces narrow, unimodal distributions depending on the value of μ, the target stem length and, therefore, is unable to capture these sequence-specific properties in the structure prediction process especially as sequence length increases.

Since the overestimation of stem and base pairing was so prevalent in our predicted structures, we next investigated if more stems could be added to any of the experimental and predicted structures. The connectivity tables of the predicted structures were parsed for base pairs and unpaired bases. If a base in the sequence was in a pair, that base was replaced with an “X” in the sequence. Then, the stem enumeration procedure was repeated (see Methods section) to determine if stems could be formed from the remaining unpaired bases. The result of this analysis for each structure is the number of possible stems that could be added to the existing structure. The results are shown in Figure . Hyperparameter set I is the only set of structures that allows for additional stems to be added but still has more structures which can not have more stems added than experimental (peak at 0 is higher in red than blue). This outcome is expected, as this parameter set favors longer stems, leaving gaps between them where shorter stems could feasibly fit. In contrast, the other hyperparameter sets predict structures that leave no room for additional stems to be added. This demonstrates that these structures are already oversaturated with stems, filling all available base-pairing opportunities, with suboptimal or incorrect pairings (low specificity, Figure S2). These results are consistent with the free energy curves shown in Figure B which demonstrated the energy will continue to decrease as you add stems. Therefore, the optimal solution according to the QUBO Hamiltonian results in structures with as many stems as possible.

5.

Distributions of the number of possible stems which could be added to each predicted (red) and experimental (blue) structure. Each plot compares the distributions for the predicted structures of the hyperparameter sets (Table S1) with the distributions for the experimental structures. Set I (left) is the only set of predicted structures where stems can be added. The distributions for Sets II–IV are delta functions at zero, indicating the predicted structures for these hyperparameters, not only overly predict base pairing, but completely saturate the structures with base pairs. Set I has structures where stems can be added, due to the nature of the constraints with these hyperparameters (see Figure S1), but still produce more structures that cannot include more stems than the experimental data. Note that no energetics are considered in this analysis and that this is the number of possible individual stems which can be added (i.e., no sequence can accommodate 200 additional stems).

These results demonstrate that our QUBO-based model consistently overpredicts the number of stems and base pairs, resulting in overly compact and physically unrealistic structures. Additionally, it selects stems from a narrow length range, failing to capture the structural heterogeneity observed in experimental RNA. Given the increasingly rugged nature of the energy landscape with deep local minima and high energy barriers shown in the previous section, further exploration of the hyperparameter phase space is both computationally intractable and unlikely to yield meaningful improvements. Instead, meaningful progress could be made with the inclusion of anticooperative terms in the Hamiltonian, terms in which the selection of one or more elements actively discourages the inclusion of others. Such additions could mitigate overestimation of base pairs and promote more diverse stem length distributions by selectively removing stems based on multibody considerations. However, a fundamental limitation of the QUBO formalism precludes this approach: the framework is restricted to pairwise interactions and cannot represent many-body constraints that are essential for modeling steric effects and volumetric exclusion. Furthermore, the pairwise terms are limited to local stem-level features, such as stem length or composition, and are insufficient to capture the global context needed to produce valid tertiary structures. As a result, the QUBO framework lacks the flexibility to model the selective behaviors and context-dependent trade-offs observed in real RNA folding.

Correlations Leading to Emergent Higher-Order Structures Will Not Occur for Biologically Relevant System Sizes

As it is not possible to explicitly incorporate multibody interactions within the QUBO framework, as discussed in the previous section, we are left to ask whether our QUBO-based Hamiltonian exhibits the necessary correlations to support the emergence of higher order structures needed for realistic RNA folding. The QUBO model shares the same mathematical form of the Hamiltonian as the Ising model for magnetic spins. The Ising model is characterized by long-range correlations under the critical temperature leading to collective behavior of the lattice. Our QUBO model specifically (eq ) is analogous to an all-to-all connected Ising lattice, in which every spin interacts with every other spin in the system. These systems are known to have a lower, but nonzero critical temperature, than the nearest-neighbor coupling model. However, the term in eq that explicitly penalizes specific pairs of bits, preventing both from taking a value of 1 (i.e., both stems being selected in the predicted structure), is a feature unique to our model and is not present in the standard Ising Hamiltonian. We will refer to this effect as pairwise exclusion (and stem overlaps in the case of RNA folding). The impact of this exclusion term on bit behavior below the critical temperature is not immediately intuitive. To investigate its effects, we initialized a toy model lattice of 50 bits, which are all-to-all connected and are given random values of k _i to represent different stem lengths, and a random set of pairs of bits are “excluded” (both bits cannot have value 1 at the same time). We vary the number of pairwise exclusions to explore the effect of increasing number of stem overlaps on correlations within the system. We allowed them to evolve using a Metropolis Algorithm according to eq below the expected critical temperature with hyperparameters as defined in Methods section. The order parameter and pair correlation functions were calculated.

We show in Figure A that the order parameter decreases with the fraction of pairwise exclusions. At zero pairwise exclusions, the structure is perfectly ordered, as expected for an all-to-all connected Ising lattice below the critical temperature. Figure B shows the average value of the pair correlation function between the bits as a function of the fraction of pairwise exclusions. Again, for zero pairwise exclusions, all bits are perfectly correlated. However, even for 5% of pairwise exclusions, the correlations are consistently close to zero indicating that even a few pairwise exclusions obstruct long-range correlations in the system entirely. As the percent of pairwise exclusions increases, an increase in correlations is also observed. However, based on the results for the order parameter with increasing pairwise exclusion, this correlation indicates that more bits are in the 0 position than the 1 position.

6.

The effect of increasing the number of pairwise exclusions in a binary optimizaiton problem on emergent long-range correlations. (A) The order parameter of a system of 50 bits as a function of increasing fraction of bits which cannot both be one. Without any pairwise exclusions, the order parameter is one and low values of the order parameter means more bits have the value of zero. (B) Correlation function for each pair of bits calculated for different fractions of pairwise exclusion. Correlations appear for no pairwise exclusions (where more bits adopt a value of one) and high pairwise exclusion fraction (where more bits are zero). Even low fractions of pairwise exclusions eliminate the ability for these systems to have emergent correlations.

There are two conditions that would allow long-range correlations, and therefore possible emergent higher order structures in our RNA systems, to occur. In one condition, there are no stem overlaps at all, which would mean that all possible stems could be selected together, and long-range patterns could emerge freely from these unconstrained combinations. In contrast, if there is high degree of overlap this would restrict which stems can coexist, drastically reducing the number of valid stem combinations. In such a constrained system, patterns could emerge as only a few combinations are allowed. Figure S4F,G shows that, while the number of stem overlaps increases with sequence length, the ratio of the total number of stem overlaps to stem pairs in RNA structures decreases exponentially as sequence length increases. Therefore, there are no sequences in the archiveII data set that have no overlaps, and only small sequences will have a sufficient number of overlaps that allow for sufficient correlations. Therefore, higher-order structures expected from experimental structures should not be expected to emerge from this model as the sequence length grows to biologically relevant scales. Crucially, this implies that such correlations, which are necessary for the emergence of higher-order structures, must be explicitly encoded in the Hamiltonian, as they will not arise spontaneously within this modeling framework. However, as demonstrated in the previous section, the QUBO formalism cannot represent the multibody correlations required to enforce the selective and cooperative behavior observed in real RNA folding. These fundamental constraints render the QUBO framework insufficient for accurately modeling biologically realistic RNA structures.

Discussion and Conclusions

We explored the predictive capabilities of embedding the RNA structure prediction problem into a QUBO framework with 2783 sequences of up to 300 bases in length from the archiveII data set. We focused on longer sequences than studied in prior works, − and elucidated fundamental issues with the QUBO problem formulation for RNA folding, which are exacerbated as the system size scales. We demonstrated that this approach yields lower predictive accuracy than both physics-based and machine learning models. A key contributor to this poor performance is the structure of the energy landscape: local minima are deep and narrow, leading to significant trapping, and even small structural differences are separated by prohibitively high energy barriers. This landscape pathology also manifests through a lack of permutation invariance, meaning that the predicted structure was strongly determined by the starting point for the optimizer in phase space. We further showed that hyperparameter optimization fails to mitigate these issues, as the ruggedness of the landscape renders phase space exploration both computationally intractable and ineffective. Compounding this, we found that our QUBO Hamiltonian predicts overly compact, physically unrealistic, and structurally homogeneous configurations. The addition of anticooperative many-body constraints would address these shortcomings; however such terms are fundamentally incompatible with the QUBO framework, which is restricted to pairwise interactions. Finally, while our QUBO Hamiltonian includes pairwise exclusion constraints to enforce physical feasibility (e.g., avoiding overlapping stems), we find that these constraints suppress long-range correlations, which are an essential feature for the formation of higher-order structural motifs. Together, these results demonstrate that the limitations are fundamental to the QUBO framework, and preclude its use for realistic RNA structure prediction at biologically relevant scales.

The question that remains is if any of these issues can be overcome. For example, the pairwise exclusion constraint preventing stem overlaps is an inherent property of the problem, which disallows emergent correlations. Therefore, explicit terms will have to be added to the Hamiltonian to promote higher-order structural motifs in predicted structures. Designing such explicit terms for a second degree polynomial is not straightforward. The Hamiltonian in this work has a penalty for pseudoknots, but our models drastically overestimate pseudoknot numbers in our predictions. This is consistent with the work by Zaborniak et al., who showed these models typically perform worse on structures without pseudoknots. There are many known RNA structures with pseudoknots, so they cannot be excluded from our model entirely. However, choosing which pseudoknots to promote or penalize depends on local and global properties of the structures. These local and global properties depend on the combination of other stems have been added to the structure. Therefore, the energetics of these terms cannot be accounted for entirely before optimization begins without trying to solve the problem by brute force at roughly atomistic resolution. QUBO algorithms, and quantum computers, do not support updating the Hamiltonian between optimization steps, and, even if they could, the computational cost of reconstructing the matrix would be enormous for larger systems. Additionally, the energetics of these high-dimensional interactions have to then be mapped to a quadratic Hamiltonian, which is yet another challenge. These same problems apply to the design of other higher-order constraints for properties other than pseudoknots. Therefore, designing terms for the Hamiltonian to promote correlations or prevent the overestimation of base pairs is very challenging, and the solution is entirely unclear.

Similarly, including extra one-body constraints will likely not help in overcoming these issues. One-body terms can only influence stem properties in our embedding of the problem. Furthermore, they can do so only in highly constrained ways, for example by specifying the total number of stems or by incorporating simple properties of individual stems, such as their length or base composition. In this work, we explicitly show that optimizing for different stem lengths can prevent overestimation of base pairs but does not improve predictive strength over existing classical models. Alevras et al. applied a one-body penalty based on the end base pair of a stem. One could imagine other one-body terms based on properties of stems that could be optimized by thorough structural studies, such as ratio of G-C to A-U base pairs, as a naive example. However, it is not obvious that such terms would sufficiently allow for emergent correlations, particularly in the absence of two-body anticooperative terms, to improve the predictive strength of the models. As an example, it may be very difficult to prevent pseudoknot overestimation based on the properties of bases or base pairs in all possible stems of a given sequence.

If changing the Hamiltonian to achieve better predictions is difficult, then maybe there are different embedding strategies that can be explored such that a Hamiltonian is simpler to construct. There are two main choices when modifying our embedding strategy. The first is to encode more structural elements in our Hamiltonian, such as hairpin loops, in addition to stems. This would allow for more constraints based on relationships between higher-order structural motifs. However, one must overcome the challenge of defining other structural motifs in a way that does not rely on (or emerge from) one or more stems forming, such as bulges, bulge loops, internal loops, or multibranch loops. Importantly, this difficulty of encoding more structural elements into a binary optimization problem remains even if the Hamiltonian is not constrained to a quadratic form. Even if solving higher-order equations can be achieved, this idea results in an increase in computational complexity of constructing the Hamiltonian and at least a quadratic increase in bits needed to represent phase space for each type of structural element considered. The inspiration for this work was to investigate the potential for hybrid classical and quantum workflows for these types of problems. As we add complexity to the Hamiltonian, the potential time speedup by solving the QUBO on a QPU could be nullified by requiring increasing amounts of classical compute time, or increased resources, preparing the Hamiltonian.

The second embedding option is to consider only base pairs or only smaller stems while promoting adjacent base pairs or stems, the latter of which was done by Alevras et al. However, this does not mitigate the issue of overly promoting base pairs due to the poor scaling of pairwise exclusion constraints, no additional anticooperative pairwise terms, and one-body terms that do not sufficiently constrain the system. For example, how does one design a term to prevent or promote a base pair from forming without information about the local (within the stem or neighboring loop) or global structure (volumetric exclusion or pseudoknots)? Additionally, only considering base pairs or smaller stems makes designing constraints for higher-order structures, like pseudoknots, which Alevras et al. did not include in their model, similarly difficult as the higher-order structural elements which directly contribute to these structural motifs are no longer directly encoded into the Hamiltonian. It is important to note that pairwise exclusions will be present regardless of whether we increase or decrease the structural resolution of our model, and therefore doing either will not allow for the emergence of higher order effects. A path forward to improve these models is therefore unclear, especially when considering the difficulty of designing a binary encoding strategy for structural elements or energetic terms to explicitly produce higher-order effects, and mapping such terms to a quadratic form.

Our analysis focuses on the energy landscape’s structure and how it is shaped by the embedding and by the quadratic form of the QUBO Hamiltonian. Although our simulations assume a noiseless process, implementations on hardware introduce platform-, optimizer-, and strategy-dependent noise whose impact on the inferred energy landscape is difficult to quantify. We focus on quantum annealers because their qubit counts are scaling most rapidly, making them the most plausible near-term platform for the input sizes considered here. Although other architectures could ultimately yield better predictions, current alternative quantum computers remain too small to impact RNA structure prediction for even moderately long sequences. An emerging focus of improvement for quantum-based RNA structure prediction is the development of hybrid classical and quantum optimization methods featuring more sophisticated quantum sampling techniques. Kumar et al. extend their previous QUBO model with a multiobjective optimization scheme, sampling-based quantum optimization, a classical training scheme for an instantaneous quantum polynomial (IQP) circuit, and classical post processing of RNA structure sampled by the quantum computer. While the extra classical postprocessing steps leads to improvements in finding the minimum energy structures, the success rate falls to 1% for a system requiring 212 qubits, corresponding to systems with approximately 14 possible stems as the Hamiltonian matrix requires a qubit for each stem pair. The common denominator between the results presented here and by Kumar et al. is the use of a QUBO Hamiltonian which will generate a free energy landscape with all of the issues outlined in our work. Therefore, our conclusions do not depend on benchmarking a particular annealer, but rather on analyzing the energy landscape itself.

The limitations we identify in our QUBO framework for RNA folding provide a potential unifying explanation for why other QUBO-based approaches have already failed in practice and why they are likely to continue to fail, especially as input sizes increase. Outside of RNA folding, other applications of QUBO to exponentially scaling problems with similar intrinsic constraints, namely pairwise exclusions, have reported lower accuracy for larger systems when compared to classical solvers. For example, Jain recently demonstrated that the QUBO formulation of the Traveling Salesman Problem (TSP) produces inaccurate solutions for even relatively small system sizes. The way in which the TSP embedding fails closely parallels what we observe in our RNA folding QUBO framework. In the TSP, a valid solution requires enforcing pairwise exclusions and ensuring that each node (city) is visited exactly once. As system size increases, the proportion of excluded paths relative to total paths diminishes, mirroring the scaling behavior we find in RNA stem overlaps. Similarly, Brubaker et al. embedded a protein docking problem into a QUBO formulation and showed that the performance of the QUBO simulated annealer rapidly degraded with system size and that larger systems led to predicted structures containing constraint violations. The constraints in the protein docking problem also include pairwise exclusion constraints similar to our RNA stem overlaps: interactions between regions of the docked and target proteins preclude these regions from interacting with other regions of the two proteins. Additionally, they found that a configuration where two nonlocal regions of the docked protein interacting with two adjacent regions of the target protein is penalized due to the unfavorable structure of the docked protein. These pairwise constraints scale similarly to those in both the TSP and our RNA structure prediction problem. Similarly to our findings, the Brubaker et al. also noted that the empirical terms of their Hamiltonian are difficult to construct, parameters are difficult to tune, and embedding this problem into a QUBO formalism makes the problems harder to solve. Our results make clear that the failures observed in other QUBO applications stem from the very same fundamental limitations we identify, and we contend that these limitations will inevitably undermine any attempt to embed exponentially scaling problem classes into a QUBO framework.

At present, this work has negative implications for the applicability of modern quantum computers to meaningfully help solve high-dimensional, exponentially scaling problems. From the hardware perspective of quantum annealers, the number of qubits required and the degree of connectivity are major hardware limitations when the systems scale exponentially and involve many pairwise interactions. Currently, D-Wave’s Advantage2 processor has 4,400 qubits and 20-way connectivity. An alternative is to utilize variational quantum algorithms which can be run on different architectures such as Quantinuum’s H2. However, these types of models suffer poor scaling and barren landscapes in the absence of initial solution guesses. , Local minima trapping is also expected to occur on quantum hardware as well and was demonstrated by Isshiki and Asari in the cases of simple equations with local minima close in value to the global minimum. As previously discussed, more advanced sampling techniques and classical post processing, while still utilizing the QUBO formalism, does not appear to mitigate the issues observed here. Ideally, hardware improvements would allow for Hamiltonians consisting of higher order polynomial or even nonpolynomial terms. With a more complex and flexible Hamiltonian, a smoother free energy landscape could be constructed, with an embedding including more structural details, allowing an annealer or other optimizer to traverse phase space effectively in exponentially scaling problems. Until quantum annealers can solve more general forms of Hamiltonians or the qubit count in other quantum computing architectures sufficiently scales, there is likely not an advantage to gain for many high-dimensional applications like TSP and RNA or protein structure prediction compared to existing classical physics-based or machine learning methods at present.

Software used to generate predicted structures, calculate energies, assess structural accuracy, and characterize structural properties like stem and base pair counts can be found here: https://github.com/andrewsb8/quvax. Scripts use to characterize phase space and experimental structures and Monte Carlo ising model calculations can be found here: https://github.com/andrewsb8/rna-paper-scripts.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcb.5c07902.

• Methods, hyperparameter selection and optimization strategies; hyperparameter values used for structure prediction (Table S1); counts of base pairs, stems, and pseudoknots in experimental and predicted structures (Table S2); sequences used for permutation invariance analyses (Table S3); energy curves of one-body terms for different hyperparameters (Figure S1); distributions of values of confusion matrix and other derived metrics for predicted structures for all hyperparameter sets (Figrue S2); F1 scores for predicted structures as a function of sequence length for each hyperparameter set (Figure S3); characterization of phase space and observed secondary structures as functions of sequence length (Figure S4); counts of base paris, stems, and pseudoknots in experimental and predicted structures (Figure S5); distributions of predicted and experimental stem lengths (Figure S6); energy and F1 score calculations for permuted inputs (Figure S7) (PDF)

∥.

D.F. and A.R. contributed equally to this work. Conceptualization: D.M.F.; Data curation: B.A.; Formal analysis: B.A. and J.P.A.; Funding acquisition: A.R.; Methodology: A.R., D.M.F., B.A.; Project administration: A.R., D.M.F.; Software: D.M.F., B.A., J.P.A.; Supervision: A.R., B.A.; Visualization: B.A., J.P.A.; Writingoriginal draft: B.A., J.P.A., D.M.F.; Writingreview and editing: A.R.

The authors declare no competing financial interest.