2014/11/26 by Jędrzej Kaniewski, Jedrzej Kaniewski, Troy Lee +4
Computer Science · Physics and Astronomy · #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #FOS: Computer and information sciences #FOS: Physical sciences #Machine Learning and Algorithms #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #cs.CC #quant-ph
paper · pdf · doi:10.48550/arxiv.1411.7280
16 pages LaTeX
arxiv created 2014/11/26 · openalex publication_date 2014/11/26 · arxiv updated 2014/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the query complexity of computing a function f:0,1n-->R+ in expectation. This requires the algorithm on input x to output a nonnegative random variable whose expectation equals f(x), using as few queries to the input x as possible. We exactly characterize both the randomized and the quantum query complexity by two polynomial degrees, the nonnegative literal degree and the sum-of-squares degree, respectively. We observe that the quantum complexity can be unboundedly smaller than the classical complexity for some functions, but can be at most polynomially smaller for functions with range 0,1. These query complexities relate to (and are motivated by) the extension complexity of polytopes. The linear extension complexity of a polytope is characterized by the randomized communication complexity of computing its slack matrix in expectation, and the semidefinite (psd) extension complexity is characterized by the analogous quantum model. Since query complexity can be used to upper bound communication complexity of related functions, we can derive some upper bounds on psd extension complexity by constructing efficient quantum query algorithms. As an example we give an exponentially-close entrywise approximation of the slack matrix of the perfect matching polytope with psd-rank only 2^n1/2+epsilon. Finally, we show there is a precise sense in which randomized/quantum query complexity in expectation corresponds to the Sherali-Adams and Lasserre hierarchies, respectively.