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Explore a cutting-edge quantum algorithm through this 40-minute conference talk that introduces Hamiltonian Decoded Quantum Interferometry (HDQI), a novel approach that combines coherent Bell measurements with the symplectic representation of the Pauli group to transform Gibbs sampling and Hamiltonian optimization into classical decoding problems. Learn how HDQI prepares purifications of density matrices ρ_P(H)∝P(H)^2 for signed Pauli Hamiltonians by solving two key tasks: decoding errors on classical codes defined by the Hamiltonian and preparing pilot states that encode anti-commutation structures. Discover how different polynomial choices enable preparation of Gibbs states at various inverse temperatures, approximate ground states, microcanonical ensembles, and other spectral filters. Examine the connection between local Hamiltonians and LDPC codes, understanding why pilot state preparation is efficient for commuting Hamiltonians but challenging for non-commuting ones, with solutions involving matrix product state representations for Hamiltonians with logarithmic-sized anti-commutation graph components. Analyze HDQI's efficiency in preparing Gibbs states for physically motivated commuting Hamiltonians including the toric code and Haah's cubic code, while exploring the development of matching classical algorithms. Investigate applications to non-commuting semiclassical spin glasses and commuting stabilizer Hamiltonians with quantum defects, where HDQI achieves Gibbs state preparation up to constant inverse-temperature thresholds using polynomial quantum resources and quasi-polynomial classical pre-processing, positioning this work as the first extension of Regev's reduction to non-abelian groups and establishing HDQI as a versatile algorithmic primitive in quantum computing.
