This is an unreviewed preprint posted to arXiv and has not yet undergone peer review. The authors study a variant of Hamiltonian learning in which a physically structured Hamiltonian—consisting of non-interacting modes plus an arbitrary spectator term—is hidden by an unknown global unitary transformation. The task is to recover the mode energies Ei without reconstructing the obfuscating unitary, a setting the authors argue is natural for materials-science applications including mean-field theory, Green's functions, and density functional theory.
Despite the apparent simplification afforded by the non-interacting structure, the paper claims that this learning problem is DQC1-hard, placing it in the complexity class associated with the "one clean qubit" model—suggesting the task is unlikely to be efficiently solvable classically in general. To address it on quantum hardware, the authors introduce SPICES (Single-Particle Inference via Contour Estimators), which pairs a Hadamard test circuit with a classical postprocessing step. The method exploits a correspondence between single-particle quasiparticle energies and the Fourier frequencies controlling zeros of a partition function evaluated at complex inverse temperature, recovering the spectral distribution in Wasserstein distance under stated assumptions.
The authors argue that SPICES is both practically simple to implement and broadly relevant, since quasiparticle spectra appear as inputs to a wide range of downstream materials calculations. If the claimed efficiency and hardness results hold up to peer scrutiny, the work would identify a concrete and well-motivated problem where near-term quantum processors could offer an advantage over classical approaches in computational materials science.