  Developing efficient quantum computing algorithms is crucial for addressing
computationally challenging problems across various fields. In this paper, we
introduce a novel quantum XZ24 algorithm, designed for efficiently computing
the eigen-energy spectra of any quantum systems. The algorithm employs an
auxiliary qubit as a control qubit to execute a pair of time-reversing
real-time evolutions of Hamiltonian $\hat{H}$ on the target qubits. The
reference state wavefunction $|\phi_0 \rangle$ is stored in target qubits. When
the control qubit (i.e., the auxiliary qubit) is in the 0 (1) state, the
$e^{-i\hat{H}t/2} (e^{i\hat{H}t/2})$ evolution operator is applied. By
combining Hadamard gates and phase gates on the auxiliary qubit, information
about $\langle \psi_0 | \cos(\hat{H}t) | \psi_0 \rangle$ can be obtained from
the output auxiliary qubit state. Theoretically, applying the Fourier
transformation to the $\langle \psi_0 | \cos(\hat{H}t) | \psi_0 \rangle$ signal
can resolve the eigen-energies of the Hamiltonian in the spectrum. We provide
theoretical analysis and numerical simulations of the algorithm, demonstrating
its advantages in computational efficiency and accuracy. Compared to existing
quantum methods, the new algorithm stands out for its remarkably low
measurement cost. For quantum systems of any complexity, only a single
auxiliary qubit needs to be measured, resulting in a measurement complexity of
$O(1)$. Moreover, this method can simultaneously obtain multiple
eigen-energies, dependent on the reference state. We anticipate that the new
algorithm will drive significant progress in quantum system simulation and
offer promising applications in quantum computing and quantum information
processing.
