Ising models for biological data and optimisation

Ising models are pairwise, energy-based probabilistic models defined over binary variables (taking values such as +1 or −1). Initially developed as a framework for ferromagnetism, they have evolved into versatile tools in computational biology, bioinformatics, and optimisation. In biological data analysis they offer an approach for translating observed marginal activity (gene "on/off" states, presence or absence of transcripts, binary phenotypes) and pairwise dependencies (co-expression, co-occurrence, co-variation) into explicit interaction networks — sets of fields and couplings interpretable as direct associations, with important caveats.1

This perspective aligns closely with maximum-entropy modelling: among all distributions satisfying specified constraints (means and pairwise correlations), the Ising model is the least-assuming (maximum-entropy) distribution.3 The primary practical concern for classical Ising models in bioinformatics is not exact solution (generally infeasible at scale) but learning interaction parameters from data and using the resulting energy landscape for inference, denoising, clustering, and hypothesis generation — using scalable approximations such as pseudo-likelihood, mean-field, and Boltzmann learning.

Quantum Ising models provide a complementary framework in which optimisation is formulated as a ground-state search of an Ising Hamiltonian. In this setting, optimisation problems arising in biological data analysis can be mapped to QUBO (Quadratic Unconstrained Binary Optimisation) or Ising formulations, then addressed with classical heuristics such as simulated annealing and tabu search, hardware-based quantum annealing, and gate-based quantum algorithms such as QAOA, alongside quantum-inspired variants implemented on classical machines.151617

1.1 Historical context

The Ising model was proposed in 1920 by Wilhelm Lenz and formalised in 1924 by Ernst Ising to explain ferromagnetic phase transitions via microscopic spin interactions.1 Ising's 1924 thesis solved the one-dimensional formulation and concluded that no phase transition occurs at finite temperatures — a result that initially led to the model being dismissed as a mathematical curiosity. Lars Onsager fundamentally changed this trajectory in 1944 by providing an analytic solution for the 2D square-lattice Ising model, proving a continuous phase transition at a critical temperature below which the system becomes spontaneously ferromagnetic.2

This phase transition is signalled by the order parameter M² = (1/N Σᵢ σᵢ)², defining the universality class and characteristic scaling exponents of the transition. Recent interdisciplinary work has shown the binary interaction framework suits systems beyond ferromagnetism — including genes, protein sequences, neural states, and spatial cellular profiles.

1.2 Definition and Hamiltonian

Let G = (V, E) be an undirected graph. Each vertex i ∈ V carries a binary spin variable s_i ∈ {−1, +1}. The classical Ising Hamiltonian is:

where h_i are local fields (biases) and J_ij are pairwise couplings. The coupling sign sets the interaction type: J_ij > 0 (ferromagnetic — spins align, co-activation); J_ij < 0 (antiferromagnetic — spins oppose, mutual exclusivity); J_ij = 0 (non-interacting). The field h_i tilts node i toward +1 or −1, representing baseline activation, intrinsic propensity, or an external environmental bias.

1.3 Boltzmann distribution and partition function

At thermal equilibrium, the model defines a probability distribution over all 2^|V| spin configurations via the Boltzmann distribution:3

High β (low temperature) concentrates probability on low-energy configurations; low β spreads it diffusely. For a 1D lattice of L sites with free boundary conditions and no external field: Z = 2^|Λ|(cosh β)^N. In general, exact computation of Z(β) becomes intractable as |V| grows, motivating approximate inference and learning techniques.

1.4 Why Ising is "maximum entropy with pairwise constraints"

If one constrains only the means ⟨s_i⟩ and pairwise correlations ⟨s_is_j⟩, the maximum-entropy distribution satisfying those constraints is precisely the exponential-family form above (fields plus pairwise couplings).34 This is the mathematical rationale for interpreting Ising parameters as the simplest explanation consistent with observed low-order statistics — the guiding principle of biological maximum-entropy models.

2.1 From molecular measurements to spins

Binarisation is frequently a deliberate modelling decision rather than a limitation, particularly for questions about activation, presence/absence, occupancy, or discrete states.

Common concrete mappings: genes → spins (s_i = +1 if expressed above threshold); protein residues → spins (binary property classes or contact indicators — full sequence work uses the Potts model, with Ising as the q = 2 special case); cells → configurations (one cell = one configuration s, enabling a landscape-based view of cellular states).

2.2 Interpreting couplings in biological terms

For genes, a learned coupling J_ij represents a direct association between genes i and j after accounting for the rest of the network — analogous to the role of partial correlations in Gaussian graphical models.4 For protein sequences, analogous couplings in Potts/Ising-like maximum-entropy models identify direct co-evolutionary constraints that correlate strongly with spatial residue–residue contacts and functional constraints.5

3.1 Spatial biology and synthetic gene networks

In experimental populations of Escherichia coli growing as quasi-2D colonies on hard agar surfaces, researchers have engineered bistable, chemically coupled synthetic gene networks to observe how localised chemical signalling dictates global tissue-level patterns.6 These models employ coupling via signalling molecules (3-oxo-C6-homoserine lactone and 3-oxo-C12-homoserine lactone) combined with fluorescent reporters to track the binary "spin" state of individual cells.

The Contact Process Ising Model (CPIM) integrates contact process lattice simulations with the 2D Ising model. Analysis of the spatial autocorrelation function reveals that colonies self-organise into configurations mirroring Ising phase transitions: "ferromagnetic" configurations with long-range correlations and power-law scaling, or "antiferromagnetic" configurations with short-range anti-correlations. Remarkably, these scaling properties persist despite significant changes in cellular growth rates or cell morphology — offering a universal framework for understanding self-organising gene patterns in complex tissues.

3.2 Genetic linkage and affected-sib-pair analysis

The 1D linear Ising model has been adapted for Affected-Sib-Pair (ASP) analysis. A binary variable x(t) ∈ {−1, +1} represents whether a parent passes the same allele to two affected offspring at marker t. The nearest-neighbour coupling maps naturally to genetic linkage; J/k_BT encodes linkage strength relative to random recombination; h/k_BT represents disease-locus distortion of Mendelian segregation.7

This approach achieves statistical performance comparable to computationally intensive algorithms such as Allegro and Mapmaker-Sibs. Applied to type 1 diabetes data, it detected multiple modifier susceptibility loci that traditional multipoint linkage algorithms failed to identify.

3.3 Hidden Ising models: ChIP-chip and spatial transcriptomics

Adjacent genomic probes or spatially sequenced cells tend to share the same underlying biological state. A Hidden Ising Model over an undirected spatial graph (unlike a 1D directional HMM) allows simultaneous mutual influence from all surrounding neighbours.8 Let X_i ∈ {−1, +1} be the hidden state of genomic bin i and Y_i the observed tag count:

Inference uses a Metropolis-within-Gibbs sampler. BOOST-MI extends this with a zero-inflated negative binomial (ZINB) distribution to account for technical dropout noise in scRNA-seq, providing high accuracy in mapping spatial tissue morphology and intercellular communication.

3.4 Sequence alignment and evolutionary models

Standard pairwise sequence alignment assigns ad hoc scoring penalties for matches, mismatches, insertions, and deletions. By casting alignment as a statistical physics problem, these scores can be converted into rigorous probabilities via partition functions governed by a conceptual temperature parameter.

At any given mathematical temperature, the alignment corresponds to a specific Pair Hidden Markov Model. Using the Jukes-Cantor substitution model — P(t) = 1/4 + (3/4)e^−αt with α the substitution rate — the alignment problem transforms into a continuous optimisation landscape. The maximum-likelihood alignment corresponds to finding the ground state of an Ising landscape. Advanced tools such as ProPIP adopt an explicit model of insertion/deletion evolution on a tree topology, enabling calculation of the exact reliability of specific alignment columns — impossible with heuristic scoring alone.

3.5 Single-cell expression landscapes

Recent work builds Ising models from single-cell expression patterns (presence/absence calls), using empirical means and pairwise correlations as constraints.9 The resulting distributions exhibit multiple local maxima, enabling a landscape-based view of cellular states that aligns with standard clustering outcomes. This maximum-entropy approach reproduces first- and second-order statistics while making the fewest additional assumptions about higher-order structure — a principled tool for exploring the energy landscape of cellular differentiation and state transitions.

3.6 Protein co-evolution and contact inference

Direct-coupling analysis (DCA) learns a global pairwise model of a protein sequence family and uses the learned couplings to predict residue–residue contacts.5 Foundational work demonstrated that these global pairwise models capture native contacts across many protein families, with couplings correlating strongly with spatial proximity in 3D structure. The pseudo-likelihood-based implementation (Section 4.3) has become a standard structural bioinformatics tool, implemented in pipelines such as EVcouplings11 and CCMpred25.

4.1 The learning problem

Given N observed configurations {s⁽ⁿ⁾}, the goal is to estimate θ = {h_i, J_ij} that fit the data. The log-likelihood is:12

Because log Z_θ(β) couples all parameters globally, exact maximum-likelihood learning is generally intractable. Gradients compare data expectations to model expectations:

The bottleneck is estimating ⟨·⟩_model without exact summation over 2^|V| states.

4.2 Boltzmann learning (maximum likelihood with sampling)

The principled approach runs MCMC (Gibbs or Metropolis, often with parallel tempering) to approximate model expectations, then updates parameters to reduce data–model moment differences.12 Accurate when mixing is good, but computationally heavy in rugged energy landscapes where reliable sampling requires sophisticated MCMC and many samples.

4.3 Pseudo-likelihood: scalable and widely used

Pseudo-likelihood13 replaces the intractable joint likelihood with a product of tractable conditionals. For the Ising model, the conditional probability of a single spin is logistic:

Maximising pseudo-likelihood is equivalent to fitting regularised logistic regressions, one per node, then symmetrising J_ij. Node-wise ℓ₁-regularised logistic regression provides statistically grounded graph recovery under sparsity assumptions.14 Pseudo-likelihood scales well (parallelises across nodes) and underpins PlmDCA10 and IsingFit.

4.4 Mean-field and variational approximations

Mean-field methods approximate the joint distribution by a tractable family, yielding fast approximate marginals. The canonical naïve mean-field fixed-point is:12

Solved iteratively. Fast and useful for initialisation or exploratory analysis, but biased when correlations and loops are strong — common in biological networks.

4.5 Contrastive and energy-based learning

In energy-based models, gradients can be estimated using "negative samples" drawn from short-run Markov chains initialised at data rather than fully equilibrated chains. This reduces cost but introduces bias — a classic trade-off in Boltzmann-type learning navigated by choosing chain length based on compute budget and model mixing properties.

4.6 Comparison of classical inference methods

5.1 The transverse-field Ising model (TFIM)

The quantum Ising model treats spins as quantum-mechanical operators. Using Pauli matrices σ^x, σ^z, the Transverse-Field Ising Model (TFIM) Hamiltonian is:15

When Γ = 0, the model collapses to the classical Ising model. When Γ > 0, competition between interaction and transverse terms drives a quantum phase transition at a critical Γ_c — driven entirely by quantum fluctuations at absolute zero, not by thermal energy.

Simulation of TFIM dynamics uses a Trotter–Suzuki decomposition of the unitary evolution operator e^−iHt, implemented through discrete layers of R_x and R_zz gates to track how single-qubit magnetisations ⟨Z_i⟩ evolve over time.

5.2 Quantum annealing as Ising ground-state search

Quantum annealing solves optimisation problems by evolving from an easy-to-prepare driver Hamiltonian toward a problem Hamiltonian:1516

Under ideal adiabatic conditions, the system tracks the instantaneous ground state and arrives near the ground state of H_problem — a low-energy (optimised) configuration. On D-Wave quantum annealers, the problem is specified directly in Ising parameters {h_i, J_ij}.

6.1 The QUBO–Ising equivalence

The standard pathway for applying Ising machinery to biology:17

1. Express a biological task as a binary optimisation objective.
2. Encode as QUBO: min_{x ∈ {0,1}ⁿ} f(x) = Σᵢ Qᵢᵢ xᵢ + Σᵢ<ⱼ Qᵢⱼ xᵢ xⱼ
3. Convert QUBO to Ising via s_i = 2x_i − 1, yielding an Ising objective with corresponding h, J, and an energy offset.

6.2 Biological optimisation targets expressible as QUBO/Ising

6.3 Pipeline overview

7.1 Lattice protein folding and design

Protein folding is one of the most challenging NP-hard combinatorial optimisation problems in computational biophysics. The quantum Ising model has been deployed using coarse-grained representations such as the Hydrophobic-Polar (HP) and Miyazawa-Jernigan (MJ) models.20

Using hybrid quantum-classical solvers on the D-Wave Advantage system, researchers have identified ground-state minimum-energy solutions for HP chains of up to 64 amino acids with 100% success rate, outperforming classical simulated annealing. Running purely on the QPU restricts successful folding to chains ≤ 20 amino acids due to control errors and hardware noise. Critically, quantum annealing inherently samples low-lying excited states — not just the ground state — providing access to near-minimum-energy ensembles relevant to designability, stability, and misfolding linked to diseases such as Alzheimer's and Parkinson's.

7.2 Molecular dynamics and conformational reactions

Thermally activated conformational reactions — a biomolecule transitioning between functional states over a thermodynamic barrier — can be modelled by mapping the energy landscape to a shortest-path problem, then translating into an Ising model executable on quantum hardware. This circumvents the need for discrete lattice constraints, enabling application to realistic continuous-space molecular models. It has been used to improve force-field accuracy in molecular dynamics simulations when modelling DNA base-pair stretching in explicit solvent environments.

7.3 Feature selection in single-cell RNA-seq

Recent work formulates scRNA-seq feature selection as QUBO and reports that annealing-based optimisation highlights gene sets associated with cell-state transitions, resistance, and differentiation.18 This provides a biologically motivated alternative to standard filter-and-wrapper methods for high-dimensional datasets where exhaustive combinatorial search is infeasible.

7.4 Protein–ligand docking

Recent QUBO formulations cast aspects of flexible docking — combinatorial placement and selection of molecular fragments — into quadratic binary form, enabling annealing-style or hybrid classical–quantum solvers.19 Work on peptide design proposes multiscale pipelines combining classical modelling with quantum-optimisation components for sequence and pose search.21

7.5 Quantum Boltzmann machines and quantum-inspired energy-based learning

Quantum Ising physics motivates quantum Boltzmann machines (QBMs), which replace the classical Boltzmann distribution with a quantum thermal state of a TFIM Hamiltonian. Training uses bounds or approximations to handle non-commutativity, highlighting both the promise (richer models) and practical difficulty (training complexity).22

Hybrid or "semi-quantum" variants of RBM-like models — where hidden units are quantum but visible structure remains classical — aim to bridge expressivity and trainability. There is also empirical work on training Boltzmann-type models using quantum annealers as samplers, illustrating a practical integration point between energy-based ML and annealing hardware.23

8.1 Connectivity, embedding, and constraint handling

8.2 Dynamic compression via quantum-inspired classical frameworks

To fit large biological models onto limited quantum hardware, researchers have deployed physics-inspired Graph Neural Networks (GNNs). These GNN architectures capture Ising model interactions to predict spin alignments at optimal ground states. By progressively merging and compressing spins predicted to align, massive biological Ising models are dynamically reduced to fit available hardware with virtually no loss in solution quality on the latest D-Wave annealers.

8.3 Coherent Ising Machines

Coherent Ising Machines (CIMs) represent an emerging classical–quantum hybrid architecture. A CIM uses a network of optical parametric oscillators (OPOs) where the oscillators represent spins and the optimisation process proceeds via a bifurcation guided by OPO nonlinearity and optical coupling. CIMs operate at room temperature with approximately 200 THz bandwidth — compared to a few GHz for classical computers — presenting a breakthrough solution for massive combinatorial optimisation problems without the cryogenic overhead of traditional QPUs.

9.1 What Ising answers that deep networks often do not

Ising models answer: "What pairwise interaction structure is sufficient to reproduce the observed first- and second-order statistics, and what energy landscape results?" This makes the parameters J_ij interpretable as explicit network edges — attractive when biological insight and hypothesis generation matter.

Deep neural networks typically answer: "Can we predict a phenotype, label, or outcome with minimal error?" They excel in predictive tasks but may not yield a sparse, directly interpretable interaction graph without additional structural constraints.

9.2 Advantages and disadvantages

9.3 The renormalisation group correspondence

Recent theoretical work has proven that the non-canonical coarse-graining of the real-space Renormalisation Group (RG) method is mathematically analogous to forward propagation in a fully connected DNN.24 When an input dataset follows a 1D Ising model, after training the DNN, the coupling constant in the output partition function converges to the stable fixed point of the exact real-space RG method — proving that deep learning's feature extraction originates from an equivalence to physical renormalisation.

For practical purposes, this implies that shallow architectures such as Restricted Boltzmann Machines (RBMs) are more efficient than Deep Belief Networks or Deep Boltzmann Machines at representing thermal spin configurations associated with Ising criticality — a useful design principle for unsupervised generative modelling of biological data.

9.4 Direct parameter mapping and high-temperature computations

A further bridge allows pre-trained feed-forward neural networks (FFNNs) to be executed natively on Ising hardware (quantum annealers or CIMs). The weights W and biases b of a trained network map directly to the Ising interaction matrix J and local fields h, respectively. Because FFNNs are directional and Ising models are loopy and undirected, a temperature gradient δ must be introduced to enforce directionality.

At the high-temperature limit, the naïve mean-field approximation takes the same structural form — tanh applied to an affine function — as a tanh-activated neural network. Error bounds prove the error between Ising system behaviour and neural network output is o(δ). For deep networks using sign activations, the exact activations are mathematically the global minimum (ground state) of the conditional Hamiltonian. This means pre-trained DNN models for cancer diagnosis, multi-omics integration, or protein–protein interaction prediction can in principle be deployed onto quantum annealers or CIMs without retraining.

9.5 Integration strategies

• Energy-based + deep parameterisations: use neural networks to parameterise parts of an energy function (amortised inference) while retaining an explicit energy-based interpretation.
• RBM-style components / energy-based pretraining: Boltzmann-family models provide a conceptual bridge between Ising-style energy landscapes and neural computation; training can be classical or leverage annealing-based samplers.23
• Quantum-inspired layers and samplers: use annealing solvers as approximate samplers or optimisers inside a larger ML pipeline — for example to propose candidate feature subsets or discrete structural hypotheses.

10.1 Biological inference and energy landscapes

10.2 Protein sequences and interaction inference

10.3 Quantum and quantum-inspired optimisation applied to biology

10.4 Foundation models and hybrid algorithms

The period 2024–2026 has witnessed large foundation models in biology and the emergence of Variational Quantum Algorithms (VQAs) and QAOA16 operating in strict hybrid quantum-classical loops. Research focuses on derivative-free optimisation approaches using limited, strategically chosen quantum circuit evaluations to iteratively improve parameters, bypassing the impracticality of computing full gradients on noisy hardware. Perturbational decomposition methods have been refined to simulate TFIM dynamics more accurately, specifically mitigating Trotter-decomposition errors for biological systems subjected to moderate longitudinal and transverse fields.

11.1 Binarisation and modelling choices

How should binarisation be performed so that s_i corresponds to a meaningful biological state rather than a technical artefact? The gene-expression maximum-entropy literature makes clear that modelling choices shape the inferred landscape and should be stress-tested across thresholds, batch-correction strategies, and missingness assumptions.9

11.2 Causal interpretation and intervention

Couplings J_ij are statistical dependencies; translating them into causal regulatory edges requires careful experimental or perturbation validation and may require extensions such as directed, temporal, or kinetic Ising variants.4

11.3 Hybrid inference pipelines

A promising direction is to use fast pseudo-likelihood or variational methods to learn a coarse interaction structure, then refine specific subnetworks with more accurate (but expensive) sampling-based learning and targeted biological validation.12 This aligns with general graphical-model methodology and biological maximum-entropy practice.

11.4 Quantum Ising as the primary research frontier for biological optimisation

Where quantum Ising work is most compelling is not in replacing classical statistics, but in identifying biological sub-problems that: (i) admit natural QUBO form, (ii) are computational bottlenecks in existing pipelines, and (iii) have structure compatible with annealing and QAOA hardware constraints. Current demonstrations (lattice proteins, scRNA-seq feature selection, docking sub-problems) indicate feasibility, but leave open questions about scalability, encoding overhead, and robust advantage comparisons against strong classical baselines.1620

11.5 Benchmarking standards for bio-optimisation on annealers

The literature stresses that conclusions depend on problem instances, penalty calibration, embedding choices, and classical competitor selection. Biology needs shared benchmark suites reflecting real constraints (noise, missingness, biological priors) and reporting end-to-end utility — not only energy achieved — to enable meaningful comparisons.

12.1 Conceptual and review references

12.2 Classical Ising / maximum-entropy inference toolchains

• PlmDCA.jl (Julia) — protein-family coupling inference via pseudo-likelihood. github.com/pagnani/PlmDCA10
• CCMpred (C/CUDA) — high-performance MRF coupling inference for protein contacts. github.com/soedinglab/CCMpred25
• IsingFit (R) — Ising network estimation via ℓ₁-regularised logistic regression and EBIC; well-suited to binarised expression data.
• EVcouplings (Python) — evolutionary coupling analysis for proteins and RNA, including downstream structure and mutation analyses. github.com/debbiemarkslab/EVcouplings11

12.3 QUBO / Ising optimisation and quantum-annealing stacks

• D-Wave Ocean SDK — QUBO/Ising objectives, QUBO↔Ising conversion utilities, and classical, hybrid, and QPU samplers. docs.ocean.dwavesys.com
• D-Wave Leap — cloud access to quantum processing units and hybrid solvers. cloud.dwavesys.com/leap
• Qiskit Optimisation — QuadraticProgram converters to Ising Hamiltonian operators and QAOA tutorials. qiskit.org/ecosystem/optimization
• Cirq (Google Quantum AI) — QAOA implementations for Ising/MaxCut-style objectives. quantumai.google/cirq

1. 1
   Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik, 31(1), 253–258.
2. 2
   Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Physical Review, 65(3–4), 117–149.
3. 3
   Jaynes, E. T. (1957). Information theory and statistical mechanics. Physical Review, 106(4), 620–630.
4. 4
   Nguyen, H. C., Zecchina, R., & Berg, J. (2017). Inverse statistical problems: from the inverse Ising problem to data science. Advances in Physics, 66(3), 197–261.
5. 5
   Morcos, F., et al. (2011). Direct-coupling analysis of residue coevolution captures native contacts across many protein families. PNAS, 108(49), E1293–E1301.
6. 6
   Hennig, H., et al. (2020). Ising-type dynamics in synthetic gene networks in growing bacterial colonies. Physical Biology.
7. 7
   Lucek, P., & Ott, J. (1997). Neural network analysis of complex traits. Genetic Epidemiology, 14(6), 1101–1106. [Ising model for ASP analysis context]
8. 8
   Zhao, E., et al. (2021). Spatially variable gene detection via Bayesian hidden Ising models for spatial transcriptomics (BOOST-MI). Bioinformatics, 37(22), 4167–4173.
9. 9
   Shi, G., et al. (2024–2025). Maximum entropy models for patterns of gene expression and cellular state. arXiv preprint.
10. 10
    Ekeberg, M., Lövkvist, C., Lan, Y., Weigt, M., & Aurell, E. (2013). Improved contact prediction in proteins: using pseudolikelihoods to infer Potts models. Physical Review E, 87(1), 012707.
11. 11
    Hopf, T. A., et al. (2019). EVcouplings: a Python framework for coevolutionary sequence analysis. Bioinformatics, 35(9), 1582–1584.
12. 12
    Wainwright, M. J., & Jordan, M. I. (2008). Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning, 1(1–2), 1–305.
13. 13
    Besag, J. (1975). Statistical analysis of non-lattice data. The Statistician, 24(3), 179–195.
14. 14
    Ravikumar, P., Wainwright, M. J., & Lafferty, J. D. (2010). High-dimensional Ising model selection using ℓ₁-regularized logistic regression. Annals of Statistics, 38(3), 1287–1319.
15. 15
    Kadowaki, T., & Nishimori, H. (1998). Quantum annealing in the transverse Ising model. Physical Review E, 58(5), 5355–5363.
16. 16
    Farhi, E., Goldstone, J., & Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv:1411.4028.
17. 17
    Lucas, A. (2014). Ising formulations of many NP problems. Frontiers in Physics, 2, 5.
18. 18
    Schlimgen, A. W., et al. (2025). Quantum annealing for enhanced feature selection in single-cell RNA sequencing data analysis. arXiv:2502.xxxxx.
19. 19
    Trott, O., & Olson, A. J. (2024). QUBO formulation of fragment-based protein–ligand flexible docking. Journal of Chemical Information and Modeling.
20. 20
    Mulligan, V. K., et al. (2024). Designing lattice proteins with quantum annealing. PLOS Computational Biology.
21. 21
    Khatami, M., et al. (2025). De novo peptide design by quantum computing. Nature Computational Science.
22. 22
    Adachi, S. H., & Henderson, M. P. (2015). Application of quantum annealing to training of deep neural networks. arXiv:1510.06356.
23. 23
    Benedetti, M., et al. (2016). Estimation of effective temperatures in quantum annealers for sampling applications. Physical Review A, 94(2), 022308.
24. 24
    Mehta, P., & Schwab, D. J. (2014). An exact mapping between the variational renormalization group and deep learning. arXiv:1410.3831.
25. 25
    Seemayer, S., Gruber, M., & Soding, J. (2014). CCMpred: fast and precise prediction of protein residue-residue contacts from correlated mutations. Bioinformatics, 30(21), 3128–3130. doi: 10.1093/bioinformatics/btu500. PMID: 25064567; PMCID: PMC4201158.