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We introduce a novel approach for estimating the spectrum of quantummany-body Hamiltonians, and more generally, of Hermitian operators, usingquantum time evolution. In our approach we are evolving a maximally mixed stateunder the Hamiltonian of interest and collecting specific time-seriesmeasurements to estimate its spectrum. We demonstrate the advantage of ourtechnique over currently used classical statistical sampling methods. Weshowcase our approach by experimentally estimating the spectral decompositionof a 2-qubit Heisenberg Hamiltonian on an IBM Quantum backend. For thispurpose, we develop a hardware-efficient decomposition that controls $n$-qubitPauli rotations against the physically closest qubit alongside expressingtwo-qubit rotations in terms of the native entangling interaction. Thissubstantially reduced the accumulation of errors from noisy two-qubitoperations in time evolution simulation protocols. We conclude by discussingthe potential impact of our work and the future directions of research itopens.

Exploiting Maximally Mixed States for Spectral Estimation by Time Evolution