  Quantum phase estimation is one of the most powerful quantum primitives. This
work proposes a new approach for the problem of multiple eigenvalue estimation:
Quantum Multiple Eigenvalue Gaussian filtered Search (QMEGS). QMEGS leverages
the Hadamard test circuit structure and only requires simple classical
postprocessing. QMEGS is the first algorithm to simultaneously satisfy the
following two properties: (1) It can achieve the Heisenberg-limited scaling
without relying on any spectral gap assumption. (2) With a positive energy gap
and additional assumptions on the initial state, QMEGS can estimate all
dominant eigenvalues to $\epsilon$ accuracy utilizing a significantly reduced
circuit depth compared to the standard quantum phase estimation algorithm. In
the most favorable scenario, the maximal runtime can be reduced to as low as
$\log(1/\epsilon)$. This implies that QMEGS serves as an efficient and
versatile approach, achieving the best-known results for both gapped and
gapless systems. Numerical results validate the efficiency of our proposed
algorithm in various regimes.
