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- W1558681514 abstract "It has long been known that in the usual black-box model, one cannot get super-polynomial quantum speedups without some promise on the inputs. In this paper, we examine certain types of promises, and show that they also cannot give rise to super-polynomial quantum speedups. We conclude that exponential quantum speedups only occur given structured promises on the input. Specifically, we show that there is a polynomial relationship of degree $12$ between $D(f)$ and $Q(f)$ for any function $f$ defined on permutations (elements of ${0,1,dots, M-1}^n$ in which each alphabet element occurs exactly once). We generalize this result to all functions $f$ defined on orbits of the group action $S_n$ (which acts on an element of ${0,1,dots, M-1}^n$ by permuting its entries). We also show that when $M$ is constant, any function $f$ defined on a symmetric set - one invariant under $S_n$ - satisfies $R(f)=O(Q(f)^{12(M-1)})$." @default.
- W1558681514 created "2016-06-24" @default.
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- W1558681514 date "2014-09-11" @default.
- W1558681514 modified "2023-09-27" @default.
- W1558681514 title "The Structure of Promises in Quantum Speedups" @default.
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