  We introduce a novel approach for estimating the spectrum of quantum
many-body Hamiltonians, and more generally, of Hermitian operators, using
quantum time evolution. In our approach we are evolving a maximally mixed state
under the Hamiltonian of interest and collecting specific time-series
measurements to estimate its spectrum. We demonstrate the advantage of our
technique over currently used classical statistical sampling methods. We
showcase our approach by experimentally estimating the spectral decomposition
of a 2-qubit Heisenberg Hamiltonian on an IBM Quantum backend. For this
purpose, we develop a hardware-efficient decomposition that controls $n$-qubit
Pauli rotations against the physically closest qubit alongside expressing
two-qubit rotations in terms of the native entangling interaction. This
substantially reduced the accumulation of errors from noisy two-qubit
operations in time evolution simulation protocols. We conclude by discussing
the potential impact of our work and the future directions of research it
opens.
