This repository studies a commuting, classically tractable sector of quantum Boltzmann machines. Its contribution is a preparation-aware representation study, not a new definition of a QBM, a new natural-gradient formalism, or a claim of quantum speedup.
- Amin et al., “Quantum Boltzmann Machine,” Physical Review X 8, 021050 (2018), introduced the QBM framework and training considerations for noncommuting thermal models.
- Patel et al., “Quantum Boltzmann machine learning of ground-state energies,” analyzed QBM energy optimization and thermal-state gradient estimation.
- Patel and Wilde, “Natural gradient and parameter estimation for quantum Boltzmann machines,” Physical Review A 112, 052421 (2025), developed thermal-state information matrices and natural-gradient estimation procedures.
- Minervini, Patel, and Wilde, “Evolved quantum Boltzmann machines,” Physical Review A 113, 032427 (2026), introduced the broader evolved-QBM ansatz and its information geometry.
- McClean et al., “Barren plateaus in quantum neural network training landscapes,” Nature Communications 9, 4812 (2018), established the standard random-circuit barren-plateau phenomenon. This repository does not claim that phenomenon.
- Chow and Liu, “Approximating discrete probability distributions with dependence trees,” IEEE Transactions on Information Theory 14, 462–467 (1968), established the forward-KL-optimal dependence-tree construction based on pairwise mutual information.
- Peyrard et al., “Exact and approximate inference in graphical models: variable elimination and beyond,” reviews variable elimination and the role of treewidth in exact inference.
- Möttönen et al., “Transformation of quantum states using uniformly controlled rotations,” provides the controlled-rotation machinery used in logical q-sample resource accounting.
The repository combines four elements:
- an exact decomposition of fully and partially aligned commuting Gibbs optimization geometry;
- matched sparse-representation controls at fixed treewidth, interaction count, and parameter count;
- a prospectively frozen weighted sparse-Ising confirmation on separately generated targets, comparing a native chain, a random target-supported tree, a maximum-weight target-supported tree, and the full target graph;
- exact logical q-sample preparation accounting for every confirmatory representation.
The maximum-weight spanning-tree algorithm itself is classical and is not claimed as new. The supported design result is empirical: retaining stronger target interactions improves finite-budget trainability over both a generic chain and a prespecified random target-supported tree while preserving width-one exact inference and q-sample preparation.
The deterministic MAXJ rule and the Chow–Liu rule answer different questions:
- Chow–Liu weights edges by pairwise mutual information and selects the tree minimizing forward information loss for a target probability distribution.
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MAXJweights target-Hamiltonian edges by$|J_{ij}|$ and is evaluated here for finite-budget optimization under a fixed Gibbs parameterization and for exact logical preparation cost.
Accordingly, MAXJ is not a Chow–Liu estimator and is not claimed to be forward-KL optimal. The later temperature-dependent tree study explicitly compares cooling-power and forward-KL criteria and records their operational disagreement.
The numerical calculations use exact classical enumeration at the studied sizes. The quantum relevance lies in thermal-ansatz design and coherent q-sample or purification preparation. See scientific claims and limitations for the precise claim hierarchy.