Estimating quantum amplitudes can be exponentially improved

• URL: http://arxiv.org/abs/2408.13721v1
• Date: Sun, 25 Aug 2024 04:35:53 GMT
• Title: Estimating quantum amplitudes can be exponentially improved
• Authors: Zhong-Xia Shang, Qi Zhao,
• Abstract summary: Estimating quantum amplitudes is a fundamental task in quantum computing. We present a novel framework for estimating quantum amplitudes by transforming pure states into their matrix forms. 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 amplitudes 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 their matrix forms and encoding them into density matrices and unitary operators. Our framework presents two specific estimation protocols, achieving the standard quantum limit $\epsilon^{-2}$ and the Heisenberg limit $\epsilon^{-1}$, respectively. Our approach significantly reduces the complexity of estimation when states exhibit specific entanglement properties. We also introduce a new technique called channel block encoding for preparing density matrices, providing optimal constructions for gate-based quantum circuits and Hamiltonian simulations. The framework yields considerable advancements contingent on circuit depth or simulation time. A minimum of superpolynomial improvement can be achieved when the depth or the time is within the range of $\mathcal{O}(\text{poly}\log(n))$. Moreover, in certain extreme cases, an exponential improvement can be realized. Based on our results, various complexity-theoretic implications are discussed.

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