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- W2884366902 abstract "The famous Johnson-Lindenstrauss lemma states that for any set of n vectors, there is a linear transformation into a space of dimension O(log n) that approximately preserves all their lengths. In fact, a Haar random unitary transformation followed by projection onto the first O(log n) coordinates followed by a scaling works as a valid transformation with high probability. In this work, we show that the Haar random unitary can be replaced by a uniformly random unitary chosen from a finite set called an approximate unitary t-design for t = O(log n). Choosing a unitary from such a design requires only polylogarithmic random bits as opposed to exponential in dimension random bits required to choose a Haar random unitary with reasonable precision. Moreover, since such unitaries can be efficiently implemented in the superpositional setting, our result can be viewed as an efficient quantum Johnson-Lindenstrauss transform akin to efficient quantum Fourier transforms widely used in earlier work on quantum algorithms. We prove our result by leveraging a method of Low for showing concentration for approximate unitary t-designs. We discuss algorithmic advantages and limitations of our result and conclude with a toy application to private information retrieval." @default.
- W2884366902 created "2018-08-03" @default.
- W2884366902 creator A5070609086 @default.
- W2884366902 date "2018-07-23" @default.
- W2884366902 modified "2023-10-04" @default.
- W2884366902 title "A quantum Johnson-Lindenstrauss lemma via unitary t-designs" @default.
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- W2884366902 doi "https://doi.org/10.48550/arxiv.1807.08779" @default.
- W2884366902 hasPublicationYear "2018" @default.
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