Local Relaxation Hierarchies for Quantum Ground State Energies: Convergence Guarantees and Message Passing Algorithms

arXiv:2609.40336 · quant-ph, cs.DC · Submitted 2026-09-30 · Read on arXiv

quant-ph, cs.DC

Submitted: 2026-09-30

Updated: 2026-09-30

Comments: 39 pages, 4 figures

Code: https://github.com/rick1924/gse-message-passing

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Hilbert Space — 24× in this paper · explained in 2 episodes
• Consistency — 23× in this paper · explained in 3 episodes
• Hypergraph — 8× in this paper · explained in 2 episodes
• Convergence Guarantees — 6× in this paper · explained in 3 episodes
• Gibbs State — 6× in this paper · explained in 2 episodes
• Embeddings — 5× in this paper · explained in 2 episodes
• Trace — 5× in this paper · explained in 2 episodes
• Effective Hamiltonian — 4× in this paper · explained in 2 episodes
• Benchmark — 3× in this paper · explained in 2 episodes
• Accuracy — 2× in this paper · explained in 1 episode
• Cluster expansion — 2× in this paper · explained in 2 episodes
• Entanglement — 2× in this paper · explained in 3 episodes
• Frustration — 2× in this paper · explained in 2 episodes
• Adaptation — 1× in this paper · explained in 1 episode
• Almost Commuting Hamiltonians — 1× in this paper · explained in 1 episode
• Clustering — 1× in this paper · explained in 2 episodes
• Combinatorial Optimization — 1× in this paper · explained in 2 episodes
• Fidelity — 1× in this paper · explained in 2 episodes
• Objective Function — 1× in this paper · explained in 2 episodes
• Semidefinite Program (SDP) — 1× in this paper · explained in 2 episodes
• Sparsity — 1× in this paper · explained in 4 episodes

Sources

• Quantum Markov Networks and Commuting Hamiltonians
• Rounding Almost Commuting Hamiltonians
• Lie-Schwinger block-diagonalization and gapped quantum chains
• Locality in Quantum Systems
• Field Theory and The Sum-of-Squares for Quantum Systems
• Perturbation Theory and the Sum of Squares
• Second order cone relaxations for quantum Max Cut
• Quantum variational embedding for ground-state energy problems: sum of squares and cluster selection
• Classical Estimation of the Free Energy and Quantum Gibbs Sampling from the Markov Entropy Decomposition

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