Abstract
The first separation between quantum polynomial time and classical bounded-error polynomial time was due to Bernstein and Vazirani in 1993. They first showed a O(1) vs. Ω(n) quantum-classical oracle separation based on the quantum Hadamard transform, and then showed how to amplify this into a n O(1) time quantum algorithm and a n Ω(logn) classical query lower bound.
We generalize both aspects of this speedup. We show that a wide class of unitary circuits (which we call dispersing circuits) can be used in place of Hadamards to obtain a O(1) vs. Ω(n) separation. The class of dispersing circuits includes all quantum Fourier transforms (including over nonabelian groups) as well as nearly all sufficiently long random circuits. Second, we give a general method for amplifying quantum-classical separations that allows us to achieve a n O(1) vs. n Ω(logn) separation from any dispersing circuit.
An Erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-540-70575-8_73
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Hallgren, S., Harrow, A.W. (2008). Superpolynomial Speedups Based on Almost Any Quantum Circuit. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_64
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DOI: https://doi.org/10.1007/978-3-540-70575-8_64
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