A Quantum Circuit Framework for Protein Ensemble-Level Energetics

The paper introduces a residue-level, gate-based quantum circuit framework for coarse-graining protein thermodynamics, moving beyond single-structure optimization toward ensemble-level characterization. Each amino acid is represented as a two-state qubit with stabilised and excited solvation states, initialised from solvation energetics. A structure-informed entanglement block of parametrised controlled gates encodes covalent bonds and non-covalent contacts across the residue network. The circuit is measured approximately 10 6 times, yielding an ensemble of binary thermodynamic microstates, from which energy distributions, residue-level statistical couplings, energetic sensitivities, and information gain relative to total free energy are computed.

The framework is demonstrated on two Trp-cage-based systems: the canonical Trp-cage miniprotein (PDB:1L2Y) and the disulfide-stabilised chimera (PDB:9GDL). Results show a structured, folding-funnel-like energy distribution for 1L2Y, while comparison with 9GDL reveals clear shifts in global energy and residue-level profiles, localising residues driving stabilisation and ensemble reorganisation. Dimer- and trimer-level couplings resolve both direct and interaction-mediated dependencies, showing the circuit captures network-level correlations.

The workflow comprises six stages: (1) input PDB structure mapping each amino acid to a qubit; (2) calculation of amino acid solvent exposure and local excitation energetics; (3) preparation of an initial quantum state from excitation propensity; (4) construction of an amino acid interaction graph; (5) calculation of qubit entanglement parameters from the interaction graph; (6) measurement of the quantum circuit to generate ensemble statistics; and (7) downstream clustering for dimer/trimer coupling analysis.

For each residue, the solvation state is calculated using Solvent Accessible Surface Area (SASA) normalised by residue-specific theoretical maximum accessible surface area values. Each residue is assigned two quantum states: a stabilised (ground) state with energy E0,r and an excited state with energy E1,r, represented relative to free transfer energies from water to bulk octanol. The intrinsic excitation gap is given by ΔGr = E1,r − E0,r, and an effective excitation gap accounts for the solvation state in the snapshot structure. The excitation propensity pr is obtained from a two-level Boltzmann partition.

The residue excitation propensities are encoded onto the quantum circuit using single-qubit RY rotations, producing a separable product state with no inter-residue correlations. The protein is then represented as a residue graph, where edges represent peptide-backbone connections and selected non-covalent contacts. A structural stiffness score Sij is calculated from backbone connectivity, disulfide bonding, hydrogen bonding, salt-bridge formation, and van der Waals packing, further modulated by frequency matching and mass compatibility to give the final coupling score Kij.

The entanglement block uses parametrised controlled rotations based on an effective local excitation potential μi for each residue and an excitation transfer flux νi→j between ordered residue pairs. The directed transfer probability ΔPi→j is mapped to controlled RY rotation gates. For pairs with weak thermodynamic drive, a symmetric IsingYY gate is used as a light entropy correction, conserving total excitation parity.

After measurement, each bit-string represents one residue-level microstate, and the total energy is computed using a diagonal residue-wise Hamiltonian. The empirical ensemble energy distribution converges to the histogram shown in the paper. The framework quantifies residue-level energetic sensitivity using conditional Kullback–Leibler divergence and directional excitation information for ordered residue pairs.

Results show that the entanglement block modifies the independent residue-state distribution, producing heterogeneous changes across the sequence. For 1L2Y, marked redistribution occurred around residues associated with the hydrophobic core, including Ile4 and Trp6. For 9GDL, pronounced changes were observed around Trp11 and the Gly16–Pro17–Ser18–Ser19 segment. The energy distribution for 1L2Y is concentrated within a relatively narrow energy range with several sharp peaks, while 9GDL exhibits a broader distribution with a wider spread of sampled energies, though at lower energies overall.

Energetic sensitivity analysis shows that in 1L2Y, Trp6 is the dominant sensitivity site, with additional contributions from Ile4, Tyr3, Leu7, Asn1, Leu2, and Arg16. In 9GDL, Trp11 is the dominant sensitivity site, supported by Tyr8, Ile9, Leu12, Arg6, Lys13, and Arg21. Pairwise information analysis identifies construct-specific communication modules: in 1L2Y, the strongest directional signals are concentrated within the Gly11–Pro12–Ser13–Ser14 segment, while in 9GDL, the corresponding module is shifted to the Gly16–Pro17–Ser18–Ser19 region.

Motif-level couplings aggregate residue-level directional effects over overlapping peptide dimers, consecutive peptide trimers, and peptide/non-covalent triads. In 1L2Y, the dimer-level map preferentially connects Gly/Pro/Ser-rich motifs with motifs containing Gln and Trp. In 9GDL, the dominant motif-level links shift towards acidic and cysteine-containing terminal motifs and the central Arg/Leu/Trp-containing region.

The discussion highlights that the framework maps a static protein structure to a coarse-grained thermodynamic ensemble, with the sampled Gibbs-energy distribution capturing intrinsic heterogeneity in the protein energy landscape. The approach avoids explicit time integration, barrier-crossing trajectories, and Markov-chain equilibration entirely, with each shot being an independent draw. The computational cost is determined by the number of residues, encoded interactions, circuit layers, and measurement shots.

Potential enhancements include incorporating multiple NMR structures, exploring variational algorithms, and executing on real quantum computers. The authors note challenges with real hardware including gate errors, decoherence, and readout bias. The residue representation is binary and two-level, which cannot resolve multiple rotameric states, continuous backbone dihedral angles, or explicit secondary-structure order parameters. The framework is validated through thermodynamic consistency arguments, showing that the circuit samples the Gibbs distribution of the structure-conditioned coarse-grained residue model under stated assumptions.

Improvements to AI Systems:

1. Structure-to-Ensemble Mapping Module: Enhance AI systems to convert a single static protein structure (PDB) into a full thermodynamic ensemble of microstates, rather than predicting a single conformation. The system can output energy distributions, residue-level excitation probabilities, and coupling strengths directly from structural input.

2. Quantum-Circuit-Aware Energy Sampling: Integrate a gate-based quantum circuit simulator as a generative model for protein states. The AI can learn to parameterize controlled rotations (RY gates) and entanglement blocks from interaction graphs, enabling efficient sampling of correlated residue states without explicit molecular dynamics or Monte Carlo trajectories.

3. Residue-Level Sensitivity and Information Analysis: Add a layer that computes conditional Kullback–Leibler divergence and directional excitation information (mutual information between ordered residue pairs) from sampled bit-strings. This allows the AI to identify key driver residues (e.g., Trp6 in 1L2Y) and communication modules (e.g., Gly-Pro-Ser segments) with quantitative confidence.

4. Transferable Coupling Score Prediction: Train the AI to predict structural stiffness scores (Sij) and coupling scores (Kij) from backbone connectivity, hydrogen bonds, salt bridges, and van der Waals packing. This enables the system to generalize to unseen proteins and predict which residue pairs will exhibit strong thermodynamic coupling.

5. Ensemble Comparison and Mutation Analysis: Enable the AI to compare two ensembles (e.g., wild-type vs. disulfide-stabilized chimera) and automatically localize residues driving shifts in global energy, distribution width, and motif-level couplings. This supports rational design by predicting which mutations or modifications alter the folding funnel.

6. Noise-Robust Quantum Readout Correction: Implement a post-processing module that corrects for readout bias and gate errors when running on real quantum hardware, using the measured histogram statistics to infer the true underlying distribution via error mitigation techniques (e.g., measurement calibration matrices).

7. Multi-Structure Aggregation: Extend the framework to accept multiple NMR or cryo-EM structures as input, allowing the AI to average over conformational heterogeneity and produce a consensus ensemble with confidence intervals on residue-level properties.

8. Variational Quantum Eigensolver Integration: Add a variational loop where the AI optimizes circuit parameters (rotation angles, entanglement strengths) to minimize the free energy of the sampled ensemble, enabling de novo prediction of thermodynamically preferred states without precomputed energetics.

What the Improved AI System Can Do:

• Predict the complete free-energy landscape of a protein from a single PDB file, including energy histograms, folding-funnel shape, and residue-specific stability contributions.

• Identify mutation hotspots and design stabilizing modifications by simulating ensemble shifts (e.g., predicting that adding a disulfide bond narrows the energy distribution and lowers global energy).

• Generate correlated residue-state samples that capture network-level dependencies (dimer/trimer couplings) for downstream analysis of allosteric communication.

• Provide interpretable, quantum-inspired uncertainty estimates for each residue’s excitation propensity, enabling confidence scoring in drug-binding or engineering contexts.

• Operate on near-term quantum hardware with built-in error correction, or simulate classically for large proteins with thousands of residues.

• Automatically compare multiple protein variants (e.g., mutants, homologs) and output a ranked list of residues driving thermodynamic differences, with statistical significance.

Proteins occupy heterogeneous free-energy landscapes in which high-entropy ensembles converge toward compact, low-energy basins with multiple sub-states. Molecular dynamics can access these landscapes at atomic resolution, but exhaustive sampling remains computationally demanding. Meanwhile, most quantum approaches target only single optimal structures, leaving full ensemble energetic heterogeneity unexplored. We introduce a residue-level, gate-based quantum circuit framework for coarse-graining protein thermodynamics. Each amino acid is represented as a two-state qubit (stabilised vs. excited solvation state) based on residue solvation energetics. A structure-informed entanglement block then encodes covalent and non-covalent contacts using parameterised controlled gates, embedding correlations across the residue-interaction network. Sampling the circuit (about 10 6 measurements) yields binary thermodynamic microstates used to compute protein energy distributions, residue-level statistical couplings, energetic sensitivities, and information gains relative to total free energy. We showcase the framework on the benchmark Trp-cage miniprotein 1L2Y (TC5b) and 9GDL, a disulfide-stabilised Trp-cage-fortified exenatide chimera. For 1L2Y, the circuit reproduces a structured, folding-funnel-like energy distribution. Comparative analysis with 9GDL reveals shifts in global energy distributions and residue-level stability profiles. Coupling and information-theoretic analyses localise residues associated with ensemble reorganisation, while multi-body couplings show the circuit resolves both direct and indirect statistical correlations. This framework expands quantum protein modelling beyond single-structure optimisation toward ensemble-level characterisation, capturing key features of rugged energy landscapes to guide protein design, mutation mapping, and allosteric pathway identification.