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Abstract: The first separation between quantum polynomial time and classicalbounded-error polynomial time was due to Bernstein and Vazirani in 1993. Theyfirst showed a O1 vs. Omegan quantum-classical oracle separation based onthe quantum Hadamard transform, and then showed how to amplify this into an^{O1} time quantum algorithm and a n^{Omegalog n} classical query lowerbound.We generalize both aspects of this speedup. We show that a wide class ofunitary circuits which we call dispersing circuits can be used in place ofHadamards to obtain a O1 vs. Omegan separation. The class of dispersingcircuits includes all quantum Fourier transforms including over nonabeliangroups as well as nearly all sufficiently long random circuits. Second, wegive a general method for amplifying quantum-classical separations that allowsus to achieve a n^{O1} vs. n^{Omegalog n} separation from any dispersingcircuit.



Author: Sean Hallgren, Aram W. Harrow

Source: https://arxiv.org/







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