Name
Abstract | ||
|---|---|---|
We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function f is computable by a 1-query quantum algorithm with error bounded by ε < 1/2 iff f can be approximated by a degree-2 polynomial with error bounded by ε' < 1/2. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy [21] and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis [1]. The proof uses Grothendieck's inequality to relate two matrix norms, with one norm corresponding to polynomial approximations and the other norm corresponding to quantum algorithms. We also show two results for polynomials of higher degree. First, there is a total Boolean function which requires [EQUATION] quantum queries but can be represented by a block-multilinear polynomial of degree [EQUATION]. Thus, in the general case (for an arbitrary number of queries), block-multilinear polynomials are not equivalent to quantum algorithms. Second, for any constant degree k, the two notions of approximation by a polynomial (the standard and the block-multilinear) are equivalent. As a consequence, we solve an open problem from [1], showing that one can estimate the value of any bounded degree-k polynomial p: {0, 1}n → [-1, 1] with [EQUATION] queries. |
Year | DOI | Venue |
|---|---|---|
2015 | 10.4230/LIPIcs.CCC.2016.25 | Conference on Computational Complexity |
Keywords | Field | DocType |
quantum algorithms,Boolean functions,approximation by polynomials,Grothendieck's inequality | Discrete mathematics,Combinatorics,Classical orthogonal polynomials,Minimal polynomial (field theory),Polynomial matrix,Orthogonal polynomials,Elementary symmetric polynomial,Discrete orthogonal polynomials,Degree of a polynomial,Mathematics,Difference polynomials | Journal |
Volume | ISSN | Citations |
abs/1511.08682 | 1868-8969 | 3 |
PageRank | References | Authors |
0.42 | 11 | 5 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Scott Aaronson | 1 | 1016 | 77.48 |
| Andris Ambainis | 2 | 2000 | 183.24 |
| Janis Iraids | 3 | 18 | 6.14 |
| mārtiņs kokainis | 4 | 3 | 0.75 |
| Juris Smotrovs | 5 | 51 | 7.35 |