⚛️ quantum physics

Logarithmic-scale variational quantum eigensolver for off-lattice protein structure prediction in continuous torsional angle space

This paper introduces a logarithmic-scale variational quantum eigensolver that enables the first all-atom, off-lattice protein structure prediction in continuous torsional space by reducing qubit requirements from linear to logarithmic, successfully demonstrating native-like structure recovery for small proteins on both simulators and real quantum hardware.

Original authors: Fabio Cumbo, Bryan Raubenolt, Varun Puram, Natalie Katzenmeyer, Jayadev Joshi, Daniel Blankenberg

Published 2026-09-03

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CC BY 4.0

Original authors: Fabio Cumbo, Bryan Raubenolt, Varun Puram, Natalie Katzenmeyer, Jayadev Joshi, Daniel Blankenberg

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Proteins are the molecular machines that keep life running, folding themselves into intricate three-dimensional shapes to perform their tasks. Predicting exactly how a string of amino acids will twist and turn into its final, functional form has been one of biology's most stubborn puzzles. For decades, scientists have relied on powerful classical computers to simulate these shapes, but the sheer number of possible ways a protein can contort is so vast that even the fastest supercomputers struggle to find the correct answer without getting lost. Recently, a new tool has entered the field: the quantum computer. Unlike traditional machines that process information in binary bits, quantum computers use quantum bits, or qubits, which can exist in multiple states at once, offering a potential shortcut through the maze of possibilities. However, early attempts to use these machines for protein folding were limited to simplified, blocky models that missed the fine details of real biology.

A team of researchers at the Cleveland Clinic has now taken a significant step forward by demonstrating a method that uses quantum computers to predict protein structures in a continuous, all-atom space, a level of detail previously thought impossible for near-term quantum hardware. Their work, published as a research article, introduces a new algorithm called the Quantum Torsion Folder. Instead of forcing proteins onto a rigid grid, this method treats the protein as a flexible chain of angles, encoding the twisting angles of the backbone and side chains directly into the phases of a quantum state. By using a logarithmic scaling strategy, the team showed that they could represent the complex geometry of a protein using exponentially fewer qubits than traditional methods require. In simulations, this approach successfully generated models of two small proteins, chignolin and Trp-cage, that closely resembled their experimentally determined shapes. When the researchers tested the method on actual quantum hardware, they recovered native-like structures, proving that the quantum signal could survive the noise of real-world devices, though the results varied depending on the specific machine used.

The core challenge in protein folding is navigating a landscape of possibilities that grows exponentially with the size of the protein. Imagine trying to find a single specific arrangement of a long, flexible rope among billions of other ways it could be tangled. Classical computers often get stuck in local valleys, finding a shape that is stable but not the correct one. The researchers addressed this by shifting the focus from spatial coordinates to the angles at which the protein chain bends. In their new framework, a quantum circuit acts as a generator, creating a complex wave of probabilities that encodes these bending angles. A classical computer then acts as an evaluator, taking these angles and building a full 3D model of the protein to check its physical stability, scoring it based on how well the atoms fit together and how much energy the structure requires. This creates a loop where the quantum computer suggests a shape, the classical computer grades it, and the quantum computer adjusts its parameters to try again, gradually refining the structure toward a stable fold.

To make this work on current quantum machines, the team had to solve a tricky problem: quantum computers cannot directly read the complex phases that hold the angle information. On a perfect simulator, these phases are visible, but on real hardware, only the probabilities of measuring certain states can be observed. The researchers developed a clever workaround for this. They mapped the observed probabilities into a cumulative distribution, effectively translating the noisy, measurable output of the quantum chip into the continuous angles needed to build the protein model. This allowed them to run the same algorithm on both high-fidelity simulators and real quantum processors, bridging the gap between theoretical design and practical execution.

The team tested their method on two well-known miniature proteins: chignolin, a small ten-residue peptide that folds into a tight hairpin shape, and Trp-cage, a slightly larger twenty-residue protein with a more complex mix of helices and loops. They ran thousands of independent simulations for each protein, using three different energy-scoring systems to evaluate the results: a custom-built function designed specifically for this task, the widely used Rosetta software, and the OpenMM physics engine. In the simulations, the method proved remarkably effective at sampling the correct shapes. For chignolin, the best models reached a high degree of accuracy, with some snapshots differing from the true experimental structure by less than one angstrom, a distance smaller than the width of a single atom. Even for the more difficult Trp-cage, the method found structures much closer to the native form than the final selected models, suggesting that the quantum computer was successfully exploring the right regions of the folding landscape.

A key insight from the study was that the final model produced by the optimization process was not always the most accurate one. The researchers found that by keeping a record of thousands of low-energy snapshots taken during the folding journey, they could often find a structure that was significantly better than the one the algorithm settled on at the very end. This suggests that the quantum computer is excellent at exploring the vast space of possibilities and finding promising candidates, but the challenge lies in selecting the single best candidate from that pool. The custom energy function performed best at guiding the system toward compact, native-like shapes, while the other scoring systems sometimes favored structures that were physically stable but geometrically further from the true fold.

To verify that this approach works on real hardware, the team executed their best simulation parameters on two different quantum processors from IBM: the ibm_cleveland and the ibm_miami. They ran the same circuit 300 times on each machine to account for the inherent randomness and noise of quantum measurements. The results were encouraging: on ibm_cleveland, the method recovered native-like structures in 88 out of 300 attempts, with the best model achieving an accuracy of 1.758 angstroms. On ibm_miami, the recovery rate was lower, with 45 successful structures out of 300, and the best model reaching 1.782 angstroms. The difference in performance highlighted that while the algorithm is robust, the specific hardware architecture, calibration, and noise characteristics of the quantum processor play a major role in the final outcome. The researchers noted that the ibm_miami jobs took significantly longer to run despite having fewer gates, indicating that factors beyond simple circuit size, such as pulse scheduling and measurement delays, affect efficiency.

The study concludes that high-resolution protein structure prediction is feasible on near-term quantum hardware, provided the problem is framed in a way that respects the machine's limitations. By converting the problem from a search for spatial coordinates into a search for torsional angles encoded in quantum phases, the team reduced the qubit requirement to a manageable level, allowing for all-atom, continuous-space modeling. While the method still faces challenges, particularly in the accuracy of energy scoring and the overhead of reading out results from noisy hardware, it establishes a scalable foundation for the future. The work demonstrates that quantum computers can move beyond coarse, simplified models to tackle the realistic, continuous geometry of biological molecules, offering a new pathway for understanding the fundamental mechanisms of life.

Technical Summary: Logarithmic-scale variational quantum eigensolver for off-lattice protein structure prediction in continuous torsional angle space

Problem Statement

Protein structure prediction (PSP) remains a computationally demanding challenge due to the exponential growth of conformational space as sequence length increases. While deep learning methods (e.g., AlphaFold) have achieved high accuracy, they rely heavily on evolutionary data and known structures, limiting their utility for novel sequences lacking homologs. Classical physics-based methods (molecular dynamics, ab initio folding) struggle with sampling timescales and force-field accuracy.

Existing quantum approaches to protein structure prediction (QPSP) have largely been constrained to coarse-grained lattice models (e.g., "alpha trace" or hydrophobic-polar models). These models discretize space and often lack the resolution to model all-atom interactions, side-chain rotamers, or realistic solvent effects. Furthermore, traditional encoding schemes where qubits map directly to spatial coordinates or lattice turns result in linear or super-linear qubit scaling, which precludes the simulation of complex proteins on near-term hardware. The field requires a method that moves beyond discrete lattices to continuous, all-atom representations while maintaining a qubit footprint compatible with current quantum processing units (QPUs).

Methodology

The authors propose Quantum Torsion Folder (QTF), a hybrid quantum-classical framework that encodes protein conformational degrees of freedom directly into the phase amplitudes of a quantum statevector.

1. Logarithmic Encoding Strategy

Unlike traditional methods that assign one qubit per degree of freedom, QTF utilizes a logarithmic scaling strategy. For a protein with $N$ torsional degrees of freedom (backbone $\phi, \psi, \omega$ and side-chain $\chi$ angles), the method requires only $n = \lceil \log_2 N \rceil$ qubits.

State Representation: The quantum state is represented as $|\psi(\theta)\rangle = \sum r_j e^{i\phi_j} |j\rangle$ .
Phase Extraction: Molecular torsion angles are derived from the gauge-fixed relative phases of the complex statevector amplitudes relative to a reference state ( $|0\rangle$ ). Specifically, the torsion angle $\tilde{\phi}_j$ is calculated as the wrapped difference between the phase of the $j$ -th amplitude and the reference amplitude.
Ansatz: The framework employs the EfficientSU2 ansatz (or an optional brickwork ansatz) to generate the parameterized quantum state. The circuit depth scales linearly with the number of torsions, while the qubit width scales logarithmically.

2. Decoding and Reconstruction

The framework utilizes two distinct decoders depending on the execution environment:

Statevector Decoder (Simulation): In classical simulations, the exact complex amplitudes are accessible. Torsion angles are extracted deterministically from the relative phases.
CDF Decoder (Hardware): On physical QPUs, statevectors are not directly observable. Instead, the framework maps the empirical Cumulative Distribution Function (CDF) of the measured computational basis probabilities to the torsion range $[-\pi, \pi]$ . This allows for hardware-compatible readout without requiring phase-sensitive tomography.
Coordinate Reconstruction: The decoded torsion angles are fed into a deterministic Natural Extension Reference Frame (NERF) algorithm (or PHEAT library) to analytically reconstruct full heavy-atom 3D Cartesian coordinates using fixed bond lengths and angles.

3. Hybrid Optimization Pipeline

The system operates as a Variational Quantum Eigensolver (VQE) loop:

Generation: The quantum circuit generates a conformation via the encoded phases.
Evaluation: A classical physics-based objective function (Hamiltonian) scores the reconstructed 3D structure. The authors implemented a customizable hybrid Hamiltonian including Lennard-Jones terms, Coulombic electrostatics, implicit solvent effects (SASA), hydrogen bonding, and steric clash penalties. They also integrated Rosetta and OpenMM (AMBER-based) energy functions for benchmarking.
Optimization: Classical optimizers (COBYLA and SLSQP) update the circuit parameters $\theta$ to minimize energy.
Curriculum: To mitigate barren plateaus and navigate the rugged energy landscape, a multi-stage optimization strategy is employed:
- Stage 1: Global hydrophobic collapse using derivative-free optimization (COBYLA) with high constraint weights.
- Stage 2: Local structural refinement using gradient-based optimization (SLSQP).
- Stage 3 (Custom Energy only): Natural relaxation where artificial constraints are tapered to allow physical potentials to dominate.

4. Ensemble and Snapshot Mining

Recognizing that a single optimization trajectory may not reach the global minimum, the study employs an ensemble approach with 400 independent replicas per protein/energy function. Crucially, the method retains up to 5,000 low-energy snapshots from each trajectory, rather than relying solely on the final endpoint. This "snapshot mining" approach allows for the identification of native-like conformations that may have been visited during optimization but discarded by the final selection criteria.

Key Results

The method was evaluated on two miniature protein benchmarks: Chignolin (5AWL, 10 residues) and Trp-cage (2JOF, 20 residues).

Simulation Results

Chignolin (5AWL): The method successfully sampled highly native-like conformations. The best retained snapshot achieved a C $\alpha$ RMSD of 0.623 Å, and the best final model reached 1.199 Å. Snapshot mining revealed that nearly all replicas (98–100% depending on the energy function) contained at least one snapshot below the 2.0 Å native-like threshold, whereas only a small fraction of final models did so.
Trp-cage (2JOF): This more complex target proved challenging. No final model or snapshot crossed the 2.0 Å threshold. However, snapshot mining improved the best observed RMSD from 3.512 Å (final model) to 2.501 Å (retained snapshot), demonstrating that the optimizer visited lower-energy basins than the final selection rule captured.
Energy Function Performance: The custom energy function produced final models with structural metrics (radius of gyration, end-to-end distance) closest to experimental values. However, all three backends (Custom, Rosetta, OpenMM) benefited significantly from snapshot mining, highlighting a disconnect between the lowest-energy structures and the lowest-RMSD structures in the sampled landscapes.

Hardware Execution Results

The authors executed 300 jobs each on ibm_cleveland (156 qubits) and ibm_miami (120 qubits) using "warm-started" parameters from the best simulation models for Chignolin.

Recovery Rate: Native-like structures (RMSD < 2.0 Å) were recovered in 88/300 jobs on ibm_cleveland and 45/300 on ibm_miami.
Best Hardware RMSD: The best structures achieved were 1.758 Å (ibm_cleveland) and 1.782 Å (ibm_miami).
Backend Variability: Despite ibm_miami having lower transpiled circuit depth and gate counts, it exhibited broader structural distributions and lower recovery rates, indicating that backend-specific factors (calibration, pulse scheduling, noise) significantly influence outcomes.
Decoder Impact: The hardware results utilized the CDF decoder, while simulations used the phase decoder. The fact that native-like structures were recovered on hardware using the CDF decoder validates the measurement-compatible mapping strategy.

Significance and Claims

The paper claims to present the first quantum algorithm for protein structure prediction in all-atom continuous space. Its primary contributions are:

Exponential Qubit Compression: By encoding torsional angles into relative phases, the method reduces the qubit requirement from $O(N)$ to $O(\log N)$ , making off-lattice simulations of small proteins feasible on near-term hardware.
Continuous Space Modeling: It moves the field away from coarse-grained lattice models toward realistic, all-atom representations with full side-chain flexibility.
Hardware Feasibility: It demonstrates that native-like conformations can be recovered on current QPUs using a measurement-compatible decoder, establishing a scalable foundation for hybrid quantum biophysics pipelines.
Sampling vs. Selection: The work highlights that the quantum generator is capable of sampling the correct conformational space, but the bottleneck lies in the classical energy function's ability to rank these structures and the selection of the final endpoint. The "snapshot mining" strategy is proposed as a necessary component to extract value from these high-dimensional searches.

The authors remain modest regarding current limitations, noting that the approach is sensitive to the choice of energy function, incurs significant computational overhead for hardware readout (shot complexity), and currently faces challenges in direct relative-phase recovery on noisy hardware. They position this work as a foundational step toward moving QPSP from coarse-grained lattices to realistic protein representations.