On a quantum-inspired kernel for classifying protein torsion angles

On a quantum-inspired kernel for1 classifying protein torsion angles2 Ashar Malik1,2,3 and David Ascher1,2,3 3 1School of Chemistry and Molecular Biosciences, The University of4 Queensland, Brisbane, Australia5 2Computational Biology and Clinical Informatics, Baker Heart and6 Diabetes Institute, Melbourne, Victoria, Australia7 3Australian Centre for Ecogenomics, The University of Queensland,8 Brisbane, Australia9 *Correspondence to Ashar J. Malik: [email protected], David B.10 Ascher: [email protected] Abstract12 Algorithms grounded in quantum principles need to demonstrate they are at least as expres-13 sive as established classical methods before any hardware advantage can be sought. Here we14 demonstrate that a quantum-inspired kernel reaches the same accuracy and Matthews cor-15 relation coefficient as carefully tuned radial basis function and degree-2 polynomial support16 vector machines when tasked with separating geometric regions in Ramachandran space.17 On a rigorously curated, balanced dataset of 10,000 torsion angle pairs derived from DSSP-18 annotated residues, the model achieves 98% accuracy with a Matthews correlation coefficient19 of 0.96, comparable to the top-performing classical models. By achieving true predictive par-20 ity on a well-characterised structural benchmark, the quantum-inspired kernel establishes21 that quantum-informed similarity measures already match classical baselines, laying a firm22 groundwork for future quantum-native bioinformatics workflows once suitable hardware be-23 comes available.24 Introduction25 Quantum computing promises to transform the life sciences [1, 2], from first-principles molec-26 ular simulation to large-scale combinatorial optimisation and high-dimensional sampling [2].27 1 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 7, 2025. ; https://doi.org/10.1101/2025.08.05.668681doi: bioRxiv preprint Two hurdles, however, stand between promise and practice. First, mapping a biological28 problem onto quantum hardware requires non-trivial data encodings and objective reformu-29 lations, often ballooning circuit depth beyond any near-term benefit [3, 4, 5]. Second, today’s30 noisy-intermediate-scale quantum (NISQ) devices suffer from decoherence and gate infidelity31 that sharply limit useful circuit depth [6, 7, 8]. In parallel, quantum-inspired algorithms,32 classical methods that mirror the algebraic structure of shallow quantum circuits [6, 9] have33 gained traction as a hardware-agnostic way to explore quantum principles.34 Among biological data science tasks, three categories are poised to benefit: classification35 (e.g., sequence analysis and genomics and, by extension, other areas such as fold prediction,36 structural phylogenetics etc.) [4, 8, 10, 11], optimisation (e.g., protein folding) [1, 2, 8, 12, 13],37 and sampling (conformational exploration of flexible macromolecules) [1, 6, 8]. Among these,38 classification is an attractive testbed because performance can be quantified empirically39 against well-calibrated classical baselines.40 Here we examine a quantum-inspired classifier for the two backbone torsion angles, phi41 (ϕ) and psi (ψ), whose joint distribution forms the Ramachandran plot. Because both angles42 lie in the bounded interval [−180 ◦, 180◦] and decades of crystallography have mapped the43 geometry exhaustively, Ramachandran space provides an almost “closed-world” benchmark44 or in other words where differences in model accuracy arise primarily from the modelling45 approach and not data sparsity.46 To enable this we use DSSP [14], which assigns per-residue secondary-structure states47 based on hydrogen-bond patterns and local geometry. In this study we do not classify48 DSSP states directly; instead, we estimate an empirical Ramachandran density from all49 DSSP-annotated residues, denoise it via thresholding and a simple post-processing step, and50 define the “allowed” support S ⊂ [−180◦, 180◦]2 as the remaining region of detected density;51 its complement is then taken as “disallowed”. We then ask whether membership in this52 denoised support can be approximated using only the backbone torsion pairs ( ϕ, ψ). Rather53 than reproducing the classic steric-clash map, this reframing casts the task as learning the54 geometric decision boundary in torsion space as a surrogate for DSSP’s structural judgement.55 Our goal is conceptual rather than accelerative: can a quantum-inspired, fidelity-inspired dot-56 product–squared kernel perform on par with strong classical kernels on this task? A positive57 answer offers an early, hardware-agnostic gauge of the practical value that quantum principles58 may bring to biological machine-learning workflows.59 In this work, each (ϕ, ψ) pair is mapped to a 12-dimensional (12D) trigonometric embed-60 ding that mirrors single-qubit rotation encodings. Similarity is then defined as the square61 of the inner product between embeddings, mimicking quantum-state fidelity as described62 in the Qiskit SDK [15]. The contribution thus lies in recasting a well-known kernel as a63 2 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 7, 2025. ; https://doi.org/10.1101/2025.08.05.668681doi: bioRxiv preprint fidelity measure ready for shallow quantum hardware for a biological problem. We bench-64 mark a support-vector machine (SVM) using this recast kernel against linear, radial-basis65 and polynomial SVMs across ten random train–test splits, reporting accuracy and Matthews66 correlation coefficient (MCC). While intentionally simple, the embedding→ kernel→ classi-67 fier pipeline generalises to other biological classification problems and can be ported directly68 to quantum circuits when suitable hardware becomes available.69 Methods70 Data Generation71 To construct a well-characterised classification dataset for torsion-angle analysis, we de-72 rived per-residue (ϕ, ψ) torsion angles from experimentally determined protein structures in73 the RCSB Protein Data Bank (PDB) [16] ( https://www.rcsb.org/). Backbone torsions74 were computed using DSSP, and a custom Python parser was used to extract (ϕ, ψ ) for75 all annotated residues; torsions for terminal residues were excluded. We then estimated a76 single empirical Ramachandran density over (ϕ, ψ) using 1◦ bins, applied a small probability77 threshold to remove low-support pixels, and used a connected-component filter to discard78 isolated artefacts. The surviving region defines the “allowed” support S ⊂ [−180◦, 180◦]2;79 its complement in [−180 ◦, 180◦]2 is “disallowed”.80 From these masks, we sampled N = 5000 points from S (label 1) and N = 5000 points81 from its complement (label 0) without replacement, yielding a balanced dataset of 10,00082 labelled torsion pairs used for all downstream training and evaluation. Angles were converted83 to radians prior to computing trigonometric features.84 Quantum-Inspired Feature Expansion85 Each (ϕ, ψ) pair is mapped to a 12-dimensional (12D) feature vector using a trigonometric86 embedding that mirrors the angle-encoding patterns of variational quantum circuits. Com-87 binations of sine and cosine terms emulate amplitude and phase modulations in single-qubit88 rotations, providing a classical analogue of quantum state preparation.89 The 12 features comprise the set {sin ϕ, cos ϕ, sin ψ, cos ψ, sin(ϕ + ψ), cos(ϕ + ψ), sin(ϕ90 − ψ), cos(ϕ − ψ), sin(ϕ ψ), cos(ϕ ψ), sin(2ϕ), cos(2ψ)}. Collectively, these features repro-91 duce much of the expressive power of shallow, entangled quantum circuits while remaining92 computationally lightweight. This compact 12D set was chosen empirically for this task and93 fixed for all experiments to aid reproducibility.94 3 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 7, 2025. ; https://doi.org/10.1101/2025.08.05.668681doi: bioRxiv preprint Kernels and Classification Models95 To benchmark this approach, we trained four support-vector-machine (SVM) configurations,96 each evaluated over ten random train–test splits (five-fold internal cross-validation).97 •Linear kernel (12D). A standard linear kernel applied to the 12D trigonometric98 embedding to assess whether the feature map alone has the ability to produce a linearly99 separable representation.100 •Polynomial kernel (12D). A degree-2 polynomial kernel was applied to the 12D101 features to capture pairwise interactions among the embedded features. For this hy-102 perparameters C, γ, and coef0 were tuned by grid search.103 •Radial-basis (RBF) kernel (2D). To establish a classical baseline in the native104 torsion space, an RBF kernel was applied directly to the raw (ϕ, ψ ) angles (no embed-105 dings).106 •Quantum-inspired kernel (12D). We define107 K(xi, xj) = ei, ej 2 , where ei and ej are the 12D trigonometric embeddings. Algebraically, K is a homo-108 geneous degree-2 polynomial kernel operating on our embedding. We describe it as109 “fidelity-inspired” to signal its connection to overlap-based similarities used in shal-110 low quantum circuits, while keeping the implementation purely classical to address111 our central question: do quantum-inspired kernels work well on biological data? A112 fidelity-exact variant can be obtained by unit-normalizing embeddings (or normalizing113 the Gram matrix), but this was not required for the present benchmark. The kernel114 matrix was precomputed and passed to SVC via kernel=‘precomputed’.115 Web Application Deployment116 To make the models accessible and interpretable, we developed a web application using117 the Flask framework. Users can upload PDB files or retrieve structures directly from the118 RCSB PDB. Backbone torsion angles are computed server-side, using DSSP, and classified119 into allowed or disallowed regions using both the RBF (baseline) and quantum-inspired120 SVM models. The application provides an interactive Mol* viewer [17] for 3D structure121 visualisation, and uses RCSB-Saguaro [18] as a feature viewer that overlays DSSP-derived122 ground truth with model predictions from both kernels. Probability maps for each model are123 4 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 7, 2025. ; https://doi.org/10.1101/2025.08.05.668681doi: bioRxiv preprint displayed over the torsion-angle space. Users can probe decision boundaries via an interactive124 sampler that draws (ϕ, ψ ) pairs from the empirical 2D Ramachandran density (thresholded125 and morphologically cleaned) and evaluates them with both classifiers; the sampled points126 and predictions can be downloaded. An API is also available to enable programmatic access.127 Results128 Torsion-Space Data Coverage129 DSSP-derived ( ϕ, ψ) angles for all RCSB PDB structures were first grouped into allowed130 and disallowed regions (Supplementary Fig. S1). Sparse, isolated pixels were removed with131 a connected-component filter to obtain a contiguous representation of conformational space.132 From the cleaned masks we then sampled an equal number of points from each class, yielding133 a balanced dataset of 10,000 residues (5,000 allowed and 5,000 disallowed; Supplementary134 Fig. S2).135 Classification Performance136 Table 1 summarises the mean test accuracy and MCC across ten random train–test splits.137 The linear kernel applied to the 12D embedding performs relatively poorly, confirming that138 the feature map alone is not linearly well separable. Introducing non-linearity with an RBF139 kernel in raw torsion space raises accuracy to roughly 96%. Both the degree-2 polynomial140 kernel on the 12D features and the quantum-inspired kernel reach the highest scores, with141 statistically indistinguishable metrics.142 Table 1: Mean ( ± SD) accuracy and MCC over ten seeds. Bold values indicate the two top-performing kernels. Kernel / Feature set Accuracy (%) MCC Linear (12D) 76.85 ± 0.92 0.537 ± 0.018 RBF (2D) 96.27 ± 0.40 0.926 ± 0.008 Poly-2 (12D) 97.93 ± 0.28 0.959 ± 0.005 Quantum-inspired (12D) 97.97 ± 0.22 0.960 ± 0.004 5 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 7, 2025. ; https://doi.org/10.1101/2025.08.05.668681doi: bioRxiv preprint Discussion143 Contemporary quantum processors remain in the noisy-intermediate-scale (NISQ) era, where144 decoherence, gate infidelity and limited qubit counts constrain circuit depth and computa-145 tional accuracy. Classical simulators offer a valuable testbed, yet their exponential memory146 demands restrict them to toy problems. In this context,quantum-inspired algorithms provide147 a pragmatic bridge: they allow exploration of the algebraic structure of shallow quantum148 circuits while running on conventional hardware, allowing workflows to be “quantum-ready”149 before fault-tolerant devices arrive.150 Throughout the paper we use “allowed” and “disallowed” in an operational sense: ( ϕ, ψ)151 pairs inside the denoised empirical support are labelled allowed, and those outside are labelled152 disallowed. In other words, the classifier designed in this work does not reproduce the classic153 steric-clash Ramachandran map; instead it asks whether two backbone torsion angles alone154 can approximate the hydrogen-bond and geometry test carried out by DSSP. Achieving155 almost 98% accuracy therefore shows that torsion angles (ϕ, ψ) capture a substantial fraction156 of the signal DSSP uses to infer the state of a residue.157 The choice of simple and fully interpretable dataset, Ramachandran torsion angles, is thus158 deliberate, and allows empirical examination of the value of a fidelity-motivated similarity159 measure without confounding data issues. Each torsion-angle ( ϕ, ψ) pair was embedded in160 a 12D trigonometric space that mirrors single-qubit rotation encodings, and similarity was161 taken as the square of the inner product between embeddings. Algebraically, this kernel is162 a degree-2 polynomial; conceptually, it is a fidelity-inspired dot-product–squared similarity163 on circuit-like features. With this reinterpretation in place, the resulting SVM matched the164 accuracy and MCC of strong classical baselines (RBF and poly-2) while outperforming a165 plain linear kernel applied to the same features. Because timing differences depend strongly166 on solver choice, SVC for the polynomial model versus a precomputed Gram matrix for the167 fidelity-inspired kernel, we emphasise predictive parity rather than speed-up: on a noise-free,168 well-characterised dataset, a quantum-inspired kernel is on par with established methods.169 To make the approach accessible, we have released a publicly available web server focused170 on Ramachandran classification. The site lets users upload or fetch protein structures,171 visualise DSSP ground truth, and compare per-residue predictions from an RBF and the172 quantum-inspired kernel side by side. By linking model output directly to experimental173 labels, the web application provides an interpretable dashboard from which to assess quantum174 and classical decision boundaries in a familiar biological setting.175 Many biological classification tasks are naturally binary (e.g., benign versus pathogenic176 variants, active versus inactive compounds, folded versus misfolded proteins). The present177 6 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 7, 2025. ; https://doi.org/10.1101/2025.08.05.668681doi: bioRxiv preprint study shows that quantum-inspired kernels are easy to implement, require no specialised178 hardware, and integrate seamlessly into existing machine-learning pipelines. We also spec-179 ulate that this benchmark has the potential to be used as a framework for developing and180 testing noise correction techniques, which will be critical for achieving reliable performance181 on true quantum hardware. As quantum technology matures, workflows already cast in182 fidelity-like kernels will be primed for direct deployment on fault-tolerant devices, poten-183 tially unlocking advantages that remain inaccessible to today’s purely classical approaches.184 Conclusion185 Quantum-inspired kernels provide a practical avenue for testing quantum principles on to-186 day’s classical hardware. Here we showed that our quantum-inspired kernel, which is alge-187 braically a degree-2 polynomial on our embedding, matches the accuracy and MCC of strong188 classical baselines when classifying allowed versus disallowed regions in Ramachandran space.189 To help the community explore this perspective, we have released an interactive web190 server that lets users upload or fetch protein structures, visualise DSSP ground truth, and191 compare predictions from a conventional RBF SVM with those of the quantum-inspired192 model side by side. This interface makes the kernel’s behaviour transparent on real structural193 data.194 Demonstrating predictive parity on a clean, well-understood benchmark and providing an195 open toolset paves the way for quantum-inspired kernels to be applied to broader biological196 tasks, such as benign versus pathogenic variant classification or active versus inactive ligand197 prediction well before large-scale quantum computers become commonplace.198 Availability199 The interactive web server is freely accessible athttps://biosig.lab.uq.edu.au/q_torsion.200 Funding201 D.B.A. was funded by the National Health and Medical Research Council grant no. GNT1174405.202 Competing Interests203 The authors declare no competing interests.204 7 .CC-BY-NC-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. 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