MQTE: A Measurement-Based Quantum Algorithm for Robust Energy Spectrum Estimation in the NISQ Era

Extracting energy spectra from quantum Hamiltonians is a fundamental task for quantum simulation, yet remains challenging on noisy intermediate-scale quantum (NISQ) devices. We propose Measured Quantum Time Evolution (MQTE), an ancilla-free algorithm that estimates energy gaps by applying real-time evolution to a reference state and measuring time-resolved probabilities via repeated projective measurements. Spectral analysis of these signals reveals oscillation frequencies corresponding to eigenvalue differences. Crucially, MQTE exhibits inherent robustness to quantum hardware noise and sampling errors: these disturbances manifest as a white-noise background, which does not distort the underlying spectral structure but rather obscures the frequency information. By increasing the number of measurement samples, the intensity of the background white noise can be suppressed, thereby recovering the original spectral content. We validate the algorithm's performance via numerical simulations on one- and two-dimensional Heisenberg models, demonstrating accurate extraction of energy gaps and resilience against both sampling and circuit-level noise. Experimental implementation on the superconducting quantum processor Tianyan-176-II further confirms the practical feasibility and noise tolerance of MQTE under real hardware conditions. This work provides a robust and scalable framework for quantum spectral estimation in the NISQ era.