The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We find tight or nearly tight bounds on the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following. kDistinctness: For any constant k, the approximate degree and quantum query complexity of the kdistinctness function is Ω(n3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of kdistinctness is O(n3/4−1/(2k+2−4)). Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n]→[n] is Ω~(n1/2). This proves a conjecture of Ambainis et al. (SODA 2016), and it implies tight lower bounds on the approximate degree and quantum query complexity of the following natural problems. kJunta testing: A tight Ω~(k1/2) lower bound for kjunta testing, answering the main open question of Ambainis et al. (SODA 2016). Statistical distance frommore »
This content will become publicly available on September 1, 2023
Geometry of Banach Spaces: A New Route Towards Position Based Cryptography
Abstract In this work we initiate the study of position based quantum cryptography (PBQC) from the perspective of geometric functional analysis and its connections with quantum games. The main question we are interested in asks for the optimal amount of entanglement that a coalition of attackers have to share in order to compromise the security of any PBQC protocol. Known upper bounds for that quantity are exponential in the size of the quantum systems manipulated in the honest implementation of the protocol. However, known lower bounds are only linear. In order to deepen the understanding of this question, here we propose a position verification (PV) protocol and find lower bounds on the resources needed to break it. The main idea behind the proof of these bounds is the understanding of cheating strategies as vector valued assignments on the Boolean hypercube. Then, the bounds follow from the understanding of some geometric properties of particular Banach spaces, their type constants. Under some regularity assumptions on the former assignment, these bounds lead to exponential lower bounds on the quantum resources employed, clarifying the question in this restricted case. Known attacks indeed satisfy the assumption we make, although we do not know how universal more »
 Publication Date:
 NSFPAR ID:
 10355659
 Journal Name:
 Communications in Mathematical Physics
 Volume:
 394
 Issue:
 2
 Page Range or eLocationID:
 625 to 678
 ISSN:
 00103616
 Sponsoring Org:
 National Science Foundation
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