Efficiently estimating eigenvalues of exponentially-large sparse Hamiltonians on quantum computers, known as quantum phase estimation, is a crucial problem in quantum computing. In this talk, I will present a new approach to quantum phase estimation specifically tailored for small and early fault-tolerant quantum computers and demonstrate the robustness of this method in different applications. The method translates the quantum phase estimation problem into a signal processing problem, utilizing the Fourier signal to identify the eigenvalue frequencies. We then employ the optimization method QCELS to fit the signal and approximate the eigenvalues. In addition to presenting the algorithm and theoretical results, I will offer an informal yet intuitive proof that elucidates why this method works and is well-suited for early fault-tolerant quantum computers. Moreover, this talk only requires some prior knowledge on the Hadamard test.
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