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- W4386876111 abstract "Quantum algorithms manipulate the amplitudes of quantum states to find solutions to computational problems. In this work, we present a framework for applying a general class of non-linear functions to the amplitudes of quantum states, with up-to an exponential improvement over the previous work. Our framework accepts a state preparation unitary (or block-encoding), specified as a quantum circuit, defining an $N$-dimensional quantum state. We then construct a diagonal block-encoding of the amplitudes of the quantum state, building on and simplifying previous work. Techniques from the QSVT literature are then used to process this block-encoding. The source of our exponential speedup comes from the quantum analog of importance sampling. We then derive new error-bounds relevant for end-to-end applications, giving the error in terms of $ell_2$-norm error. We demonstrate the power of this framework with four key applications. First, our algorithm can apply the important function $tanh(x)$ to the amplitudes of an arbitrary quantum state with at most an $ell_2$-norm error of $epsilon$, with worst-case query complexity of $O(log(N/epsilon))$, in comparison to the $O(sqrt{N}log(N/epsilon))$ of prior work. Second, we present an algorithm solving a new formulation of maximum finding in the unitary input model. Third, we prove efficient end-to-end complexities in applying a number of common non-linear functions to arbitrary quantum states. Finally, we generalize and unify existing quantum arithmetic-free state-preparation techniques. Our work provides an important and efficient algorithmic building block with potentially numerous applications in areas such as optimization, state preparation, quantum chemistry, and machine learning." @default.
- W4386876111 created "2023-09-20" @default.
- W4386876111 creator A5003208881 @default.
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- W4386876111 date "2023-09-18" @default.
- W4386876111 modified "2023-09-27" @default.
- W4386876111 title "Non-Linear Transformations of Quantum Amplitudes: Exponential Improvement, Generalization, and Applications" @default.
- W4386876111 doi "https://doi.org/10.48550/arxiv.2309.09839" @default.
- W4386876111 hasPublicationYear "2023" @default.
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