  Energy spectroscopy is a powerful tool with diverse applications across
various disciplines. The advent of programmable digital quantum simulators
opens new possibilities for conducting spectroscopy on various models using a
single device. Variational quantum-classical algorithms have emerged as a
promising approach for achieving such tasks on near-term quantum simulators,
despite facing significant quantum and classical resource overheads. Here, we
experimentally demonstrate multi-level variational spectroscopy for fundamental
many-body Hamiltonians using a superconducting programmable digital quantum
simulator. By exploiting symmetries, we effectively reduce circuit depth and
optimization parameters allowing us to go beyond the ground state. Combined
with the subspace search method, we achieve full spectroscopy for a 4-qubit
Heisenberg spin chain, yielding an average deviation of 0.13 between
experimental and theoretical energies, assuming unity coupling strength. Our
method, when extended to 8-qubit Heisenberg and transverse-field Ising
Hamiltonians, successfully determines the three lowest energy levels. In
achieving the above, we introduce a circuit-agnostic waveform compilation
method that enhances the robustness of our simulator against signal crosstalk.
Our study highlights symmetry-assisted resource efficiency in variational
quantum algorithms and lays the foundation for practical spectroscopy on
near-term quantum simulators, with potential applications in quantum chemistry
and condensed matter physics.
