Estimating quantum amplitudes can be exponentially improved

• URL: http://arxiv.org/abs/2408.13721v2
• Date: Sun, 08 Dec 2024 04:21:21 GMT
• Title: Estimating quantum amplitudes can be exponentially improved
• Authors: Zhong-Xia Shang, Qi Zhao,
• Abstract summary: Estimating quantum amplitude (the overlap between two quantum states) is a fundamental task in quantum computing.<n>We present a novel algorithmic framework for estimating quantum amplitudes by transforming pure states into matrix forms.<n>Our framework achieves the standard quantum limit $epsilon-2$ and the Heisenberg limit $epsilon-1$, respectively.
• Score: 11.282486674587236
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Estimating quantum amplitude (the overlap between two quantum states) is a fundamental task in quantum computing and serves as a core subroutine in numerous quantum algorithms. In this work, we present a novel algorithmic framework for estimating quantum amplitudes by transforming pure states into matrix forms and encoding them into non-diagonal blocks of density operators and diagonal blocks of unitary operators. Our framework presents two specific estimation protocols, achieving the standard quantum limit $\epsilon^{-2}$ and the Heisenberg limit $\epsilon^{-1}$, respectively. Whenever one quantum state is prepared by a $\mathit{o}(n)$-depth quantum circuit and the other has a large entanglement under a certain bi-partition, our algorithm can give exponential improvement over the direct Hadamard test and amplitude estimation algorithm for both query complexity and gate complexity. The gate complexity reduction comes from a new technique called channel block encoding. This technique provides a systematical and efficient way to embed the matrix form of a pure state into a density operator.

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