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arxiv: 2405.06657 · v2 · submitted 2024-04-19 · 🧬 q-bio.BM · cs.CE· cs.ET

Molecular Docking via Weighted Subgraph Isomorphism on Quantum Annealers

Emanuele Triuzzi , Riccardo Mengoni , Francesco Micucci , Domenico Bonanni , Daniele Ottaviani , Andrea Beccari , Gianluca Palermo This is my paper

Pith reviewed 2026-05-24 02:27 UTC · model grok-4.3

classification 🧬 q-bio.BM cs.CEcs.ET
keywords molecular dockingquantum annealingsubgraph isomorphismQUBOdrug discoveryprotein-ligand interactionsimulated annealinggraph matching
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The pith

Molecular docking is cast as weighted subgraph isomorphism between a ligand graph and a protein pocket grid, then solved as QUBO on quantum annealers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates the pose search step of molecular docking as a graph matching task. A ligand is represented as a graph that encodes its three-dimensional geometry together with internal flexibility, while the binding pocket is discretized into a weighted spatial grid. This matching problem is converted into a QUBO that can be submitted directly to a quantum annealer. The same QUBO is also solved with classical simulated annealing so that the two approaches can be compared on identical instances. A reader would care because docking is a routine but expensive calculation in early-stage drug design, and an annealing formulation could eventually let specialized hardware handle larger or more flexible ligands.

Core claim

The authors formulate molecular docking as a weighted subgraph isomorphism problem between the ligand graph (encoding geometry and flexibility) and the weighted spatial grid of the protein pocket, encode the resulting matching task as a QUBO, solve it on quantum annealers, and compare the outcomes with those obtained from classical simulated annealing on the same instances.

What carries the argument

Weighted subgraph isomorphism between the ligand graph and the pocket grid, mapped to a QUBO for annealing solvers.

If this is right

  • Quantum annealers become applicable to the pose-search phase of docking without intermediate classical preprocessing steps.
  • Ligand flexibility is handled directly by the graph representation rather than by enumerating discrete conformers.
  • Energetic preferences of the pocket are incorporated through edge weights in the grid, allowing the QUBO to reflect binding affinity contributions.
  • Performance can be benchmarked on identical QUBO instances against classical simulated annealing to quantify any quantum advantage.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If hardware noise and embedding overhead decrease, the same graph formulation could be applied to larger pockets or more flexible ligands than current classical exhaustive searches allow.
  • The approach could be combined with downstream classical scoring functions that refine the ranked poses returned by the annealer.
  • Similar graph-isomorphism encodings might be tested on other geometry-constrained problems in computational chemistry, such as protein-protein interface prediction.

Load-bearing premise

The ligand graph and the weighted pocket grid together contain all the geometric and energetic information required to recover correct binding poses.

What would settle it

On a ligand-protein pair with a known crystal structure, the lowest-energy configuration returned by the annealer lies more than 2 Å RMSD from the experimental pose.

Figures

Figures reproduced from arXiv: 2405.06657 by Andrea Beccari, Daniele Ottaviani, Domenico Bonanni, Emanuele Triuzzi, Francesco Micucci, Gianluca Palermo, Riccardo Mengoni.

Figure 1
Figure 1. Figure 1: Complete workflow: the approach addresses the SC search via QA which outputs a sample of valid configurations. The pose filter, in the BA evaluation, selects the most promising poses based on chemical score. The basic idea behind this approach is to consider interesting docking points within the pocket which identify an active region of the pocket itself. The number of pocket probes depends on the pocket s… view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the functions A(t) (blue line) and B(t) (light blue line) defining the annealing schedule. 3.1. D-Wave Quantum Annealer D-Wave Systems is nowadays the world’s leading producer of quantum annealing devices providing access to their machines via the cloud. The current most advanced QA hardware is a 5000 qubit QPU named D-Wave Advantage and a 2048-qubit QPU called D-Wave 2000Q. D-Wave devices are able… view at source ↗
Figure 3
Figure 3. Figure 3: From the molecular pocket we select the pocket points that are used to construct the 3D space grid inside the pocket. 4.2. From Ligand to molecular graph In the pre-processing phase, several simplifications are applied to ligands’ chemical and geometric properties to align them with the constraints of the current Quantum Computer, specifically in terms of qubits and their interconnections [PITH_FULL_IMAGE… view at source ↗
Figure 4
Figure 4. Figure 4: Three different pre-processing approximations: No fragment simplification, Center of mass simplification, and Internal Fragments removal Now that the ligand has undergone such simplification, information about the geometrical properties of the molecule i.e. connectivity between atoms, rotatable bonds, bonds length, and values of fixed angles are encoded into a weighted graph Gmol = {i, ei,j , wi,j}. As a s… view at source ↗
Figure 5
Figure 5. Figure 5: Example of graph construction. On the left, the original molecule obtained after preprocessing, rotatable bonds are identified with arrows. On the right, the graph structure obtained: e bond i,j , e angle i,j and e dih i,j are respectively depicted in black, blue and red in the figure. 4.3. Weighted Sub-graph Isomorphism The objective of the Molecular Docking problem can be expressed as the optimal matchin… view at source ↗
Figure 6
Figure 6. Figure 6: Number of linear terms in the QUBO for a pocket-size with Ngrid equal to 50 (left) and 100 (right), increasing ligand size, i.e. number of nodes Nmol in Gmol [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Number of quadratic terms in the QUBO for a pocket-size with Ngrid equal to 50 (left) and 100 (right), increasing ligand size, i.e. number of nodes Nmol in Gmol Given that the formulation of the problem is already in QUBO form, we avoid overhead in the generation of the QUBO, often due to the presence of high-order terms that need to be converted to quadratic and linear terms. Creation time for QUBOs is re… view at source ↗
Figure 8
Figure 8. Figure 8: QUBO creation time (in milliseconds) for ligand size of Nmol equal to 5 (left) and 10 (right) increasing pocket size Ngrid = 20, 50, 100 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Number of physical qubits obtained after the embedding for ligand size equal to 5 (left) and 10 (right) increasing the size of the pocket space grid. Colors indicate qubits obtained after embedding into D-Wave 2000Q, Advantage compared to the logical qubits before embedding. Embedding in D-Wave 2000Q is limited due to the size of the device which is able to solve problems up to 100 logical qubits and pocke… view at source ↗
Figure 10
Figure 10. Figure 10: Average number of physical qubits in a chain for ligand size equal to 5 (left) and 10 (right) increasing the size of the pocket space grid. Colors indicate qubits obtained after embedding into D-Wave 2000Q and Advantage. Embedding in D-Wave 2000Q is limited due to the size of the device [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Distribution of solutions in the output sample with respect to ABD (left) and related RMSD (right) obtained from Advantage, DW 2000Q, and Simulated Annealing (SA). The Total pairwise distance in the plot indicates the value of the optimization term Opt. The plots in Fig.11 show the distribution of solutions in the output sample with respect to the ABD and the related RMSD values obtained from Advantage, D… view at source ↗
Figure 12
Figure 12. Figure 12: Time to solution (in milliseconds) for solutions found below the three threshold values. (Recall that for TTS, the lower the better) For what concerns the TTS, solutions below the three thresholds were considered. The SA is able to find good solutions at very low TTS, if compared to the TTS of the quantum solvers. Anyway, the gap between QPUs and SA is largely reduced when looking for solutions below the … view at source ↗
Figure 13
Figure 13. Figure 13: Distribution of solutions (left) in the output sample and related RMSD (right) obtained from Advantage, DW 2000Q and Simulated Annealing (SA) [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Time to solution (in milliseconds) for solutions found below the three threshold values. (Recall that for TTS, the lower the better) [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
read the original abstract

Molecular docking is an essential step in the drug discovery process involving the detection of three-dimensional poses of a ligand inside the active site of the protein. In this paper, we address the Molecular Docking search phase by formulating the problem in QUBO terms, suitable for an annealing approach. We propose a problem formulation as a weighted subgraph isomorphism between the ligand graph and the grid of the target protein pocket. In particular, we applied a graph representation to the ligand embedding all the geometrical properties of the molecule including its flexibility, and we created a weighted spatial grid to the 3D space region inside the pocket. Results and performance obtained with quantum annealers are compared with classical simulated annealing solvers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript claims that the molecular docking search phase can be reformulated as a weighted subgraph isomorphism between a ligand graph (encoding 3D geometry and torsional flexibility) and a weighted spatial grid discretizing the protein pocket region; the resulting problem is cast as a QUBO and solved on quantum annealers, with performance compared against classical simulated annealing.

Significance. If the graph-plus-grid encoding can be shown to reproduce physically relevant binding poses and scores without post-hoc corrections, the work would constitute a concrete demonstration of quantum annealing applied to a high-dimensional combinatorial search problem in structural biology. The direct comparison to simulated annealing supplies a necessary classical baseline.

major comments (3)
  1. [Abstract / formulation] Abstract and formulation section: the central claim that the weighted subgraph isomorphism yields poses whose ranking matches physical binding affinity rests on the unshown assertion that a single set of edge weights on the ligand graph and pocket grid can encode all relevant energetic contributions. No derivation is supplied showing how hydrogen-bond, electrostatic, or desolvation terms are projected onto these weights; if they are omitted or approximated only geometrically, the QUBO optimum need not correspond to a low-RMSD or high-affinity pose.
  2. [Methods] Methods: the description of the ligand graph states that it 'embeds all the geometrical properties including its flexibility,' yet no explicit construction (vertex/edge attributes, distance or angle thresholds, or torsional variable encoding) is provided. Without this, it is impossible to verify that the resulting QUBO Hamiltonian is faithful to the original docking objective.
  3. [Results] Results: performance is compared to classical simulated annealing, but no quantitative metrics (RMSD distributions, enrichment factors, or success rates on a standard benchmark set such as PDBbind) are reported in the abstract or visible text. The absence of such validation data leaves the practical utility of the QUBO mapping untested.
minor comments (1)
  1. [Methods] Notation for the weighted grid and isomorphism objective should be introduced with explicit symbols rather than prose descriptions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below, with planned revisions to improve clarity on the scope and details of the formulation.

read point-by-point responses

  1. Referee: [Abstract / formulation] Abstract and formulation section: the central claim that the weighted subgraph isomorphism yields poses whose ranking matches physical binding affinity rests on the unshown assertion that a single set of edge weights on the ligand graph and pocket grid can encode all relevant energetic contributions. No derivation is supplied showing how hydrogen-bond, electrostatic, or desolvation terms are projected onto these weights; if they are omitted or approximated only geometrically, the QUBO optimum need not correspond to a low-RMSD or high-affinity pose.

    Authors: We agree that no derivation mapping full physical terms (hydrogen bonds, electrostatics, desolvation) onto the edge weights is provided. The formulation encodes geometric compatibility via distance-based weights between ligand atoms and grid points. We will revise the abstract and formulation section to explicitly qualify the approach as a geometric matching approximation and remove or qualify any implication that the QUBO optimum directly ranks by physical binding affinity. revision: yes

  2. Referee: [Methods] Methods: the description of the ligand graph states that it 'embeds all the geometrical properties including its flexibility,' yet no explicit construction (vertex/edge attributes, distance or angle thresholds, or torsional variable encoding) is provided. Without this, it is impossible to verify that the resulting QUBO Hamiltonian is faithful to the original docking objective.

    Authors: We acknowledge that the explicit construction details for the ligand graph (vertex/edge attributes, thresholds, and torsional encoding) are not supplied. In the revised Methods section we will add the precise definitions of vertices and edges, the distance and angle criteria used to define edges, and the binary variables introduced to represent torsional degrees of freedom. revision: yes

  3. Referee: [Results] Results: performance is compared to classical simulated annealing, but no quantitative metrics (RMSD distributions, enrichment factors, or success rates on a standard benchmark set such as PDBbind) are reported in the abstract or visible text. The absence of such validation data leaves the practical utility of the QUBO mapping untested.

    Authors: The reported results emphasize runtime and solution-quality comparisons on the QUBO objective itself rather than end-to-end docking benchmarks. We will expand the Results section to include RMSD values for the poses obtained on the example systems already presented and add an explicit limitations paragraph noting the lack of PDBbind-scale enrichment statistics. revision: partial

Circularity Check

0 steps flagged

No circularity: direct encoding presented without self-referential reductions

full rationale

The paper formulates molecular docking as a weighted subgraph isomorphism QUBO between a ligand graph and a protein pocket grid, then compares quantum and classical annealing solvers. No equations, fitted parameters, or predictions are shown that reduce by construction to inputs; the mapping is introduced as an explicit construction rather than a derived result. No self-citations, uniqueness theorems, or ansatzes appear in the provided text as load-bearing steps. The central claim is the encoding itself, which stands as an independent proposal without circular reduction to prior outputs or definitions within the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated premise that the chosen graph and grid weights preserve docking physics.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Physically-Informed Subgraph Isomorphism Approach to Molecular Docking Using Quantum Annealers

    cs.ET 2026-04 unverdicted novelty 5.0

    A novel QUBO formulation for quantum-annealer molecular docking adds physicochemical interaction terms to a prior geometric subgraph-isomorphism approach and reports improved accuracy on D-Wave devices.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 1 Pith paper

  1. [1]

    Morris and Marguerita Lim-Wilby.Molecular Docking, pages 365–382

    Garrett M. Morris and Marguerita Lim-Wilby.Molecular Docking, pages 365–382. Humana Press, Totowa, NJ, 2008

    work page 2008

  2. [2]

    Protein-liganddocking: Current status and future challenges

    SergioFilipeSousa, PedroAlexandrinoFernandes, andMariaJoaoRamos. Protein-liganddocking: Current status and future challenges. Proteins:Structure, Function, and Bioinformatics , 65(1):15–26, 2006

    work page 2006

  3. [3]

    Computationalmethodsforbiomoleculardocking

    ThomasLengauerandMatthiasRarey. Computationalmethodsforbiomoleculardocking. Current Opinion in Structural Biology, 6(3):402–406, 1996

    work page 1996

  4. [4]

    Paul C. D. Hawkins, A. Geoffrey Skillman, and Anthony Nicholls. Comparison of shape-matching and docking as virtual screening tools.Journal of Medicinal Chemistry, 50(1):74–82, 01 2007

    work page 2007

  5. [5]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang.Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010. Molecular Docking via Weighted Subgraph Isomorphism on Quantum Annealers 19

    work page 2010

  6. [6]

    Gabb, Richard M

    Henry A. Gabb, Richard M. Jackson, and Michael J.E. Sternberg. Modelling protein docking using shape complementarity, electrostatics and biochemical information11edited by j. thornton. Journal of Molecular Biology, 272(1):106–120, 1997

    work page 1997

  7. [7]

    Caviar: a method for automatic cavity detection, description and decomposition into subcavities

    Jean-Rémy Marchand, Bernard Pirard, Peter Ertl, and Finton Sirockin. Caviar: a method for automatic cavity detection, description and decomposition into subcavities. Journal of Computer-Aided Molecular Design, 35(6):737–750, 2021

    work page 2021

  8. [8]

    Patrick Brady and Pieter F.W

    G. Patrick Brady and Pieter F.W. Stouten. Fast prediction and visualization of protein binding pockets with pass. Journal of Computer-Aided Molecular Design, 14(4):383–401, May 2000

    work page 2000

  9. [9]

    Roll: a new algorithm for the detection of protein pockets and cavities with a rolling probe sphere.Bioinformatics, 26(1):46–52, 10 2009

    Jian Yu, Yong Zhou, Isao Tanaka, and Min Yao. Roll: a new algorithm for the detection of protein pockets and cavities with a rolling probe sphere.Bioinformatics, 26(1):46–52, 10 2009

    work page 2009

  10. [10]

    Quantum molecular unfolding

    Kevin Mato, Riccardo Mengoni, Daniele Ottaviani, and Gianluca Palermo. Quantum molecular unfolding. Quantum Science and Technology, 7(3):035020, jun 2022

    work page 2022

  11. [11]

    Encoding molecular docking for quantum computers.Journal of Chemical Theory and Computation, 19(24):9018–9024, 2023

    Jinyin Zha, Jiaqi Su, Tiange Li, Chongyu Cao, Yin Ma, Hai Wei, Zhiguo Huang, Ling Qian, Kai Wen, and Jian Zhang. Encoding molecular docking for quantum computers.Journal of Chemical Theory and Computation, 19(24):9018–9024, 2023. PMID: 38090816

    work page 2023

  12. [12]

    Molecular docking with gaussian boson sampling.Science Advances, 6(23):eaax1950, 2020

    Leonardo Banchi, Mark Fingerhuth, Tomas Babej, Christopher Ing, and Juan Miguel Arrazola. Molecular docking with gaussian boson sampling.Science Advances, 6(23):eaax1950, 2020

    work page 2020

  13. [13]

    Molecular docking via quantum approximate optimization algorithm, 2024

    Qi-Ming Ding, Yi-Ming Huang, and Xiao Yuan. Molecular docking via quantum approximate optimization algorithm, 2024

    work page 2024

  14. [14]

    A quantum algorithm for the sub-graph isomorphism problem, 2022

    Nicola Mariella and Andrea Simonetto. A quantum algorithm for the sub-graph isomorphism problem, 2022

    work page 2022

  15. [15]

    D.J.J.Marchand, M.Noori, A.Roberts, G.Rosenberg, B.Woods, U.Yildiz, M.Coons, D.Devore, and P. Margl. A variable neighbourhood descent heuristic for conformational search using a quantum annealer. Scientific Reports, 9(1):13708, Sep 2019

    work page 2019

  16. [16]

    Barkoutsos, and Ivano Tavernelli

    Stefano Mensa, Emre Sahin, Francesco Tacchino, Panagiotis Kl. Barkoutsos, and Ivano Tavernelli. Quantum machine learning framework for virtual screening in drug discovery: a prospective quantum advantage. CoRR, abs/2204.04017, 2022

    work page arXiv 2022

  17. [17]

    Coarse-grained lattice protein folding on a quantum annealer, 2018

    Tomas Babej, Christopher Ing, and Mark Fingerhuth. Coarse-grained lattice protein folding on a quantum annealer, 2018

    work page 2018

  18. [18]

    Catherine C. McGeoch. Adiabatic Quantum Computation and Quantum Annealing: Theory and Practice. 2014

    work page 2014

  19. [19]

    Siteferret: Beyond simple pocket identification in proteins

    Luca Gagliardi and Walter Rocchia. Siteferret: Beyond simple pocket identification in proteins. Journal of Chemical Theory and Computation, 19(15):5242–5259, 2023. PMID: 37470784

    work page 2023

  20. [20]

    The pdbbind database: Methodologies and updates.Journal of Medicinal Chemistry, 48(12):4111–4119, 2005

    Renxiao Wang, Xueliang Fang, Yipin Lu, Chao-Yie Yang, and Shaomeng Wang. The pdbbind database: Methodologies and updates.Journal of Medicinal Chemistry, 48(12):4111–4119, 2005

    work page 2005