



@article {2880,
	title = {Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy},
	year = {2021},
	month = {10/9/2021},
	abstract = {<p>We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that deciding whether the (finite-dimensional) quantum value of a nonlocal game is 1 or at most 12 is complete for the class Σ1 (i.e., RE). A result of Slofstra implies that deciding whether the commuting operator value of a nonlocal game is equal to 1 is complete for the class Π1 (i.e., coRE). We prove that deciding whether the quantum value of a two-player nonlocal game is exactly equal to 1 is complete for Π2; this class is in the second level of the arithmetical hierarchy and corresponds to formulas of the form \&quot;\&forall;x\&exist;yϕ(x,y)\&quot;. This shows that exactly computing the quantum value is strictly harder than approximating it, and also strictly harder than computing the commuting operator value (either exactly or approximately). We explain how results about the complexity of nonlocal games all follow in a unified manner from a technique known as compression. At the core of our Π2-completeness result is a new \&quot;gapless\&quot; compression theorem that holds for both quantum and commuting operator strategies. Our compression theorem yields as a byproduct an alternative proof of Slofstra\&$\#$39;s result that the set of quantum correlations is not closed. We also show how a \&quot;gap-preserving\&quot; compression theorem for commuting operator strategies would imply that approximating the commuting operator value is complete for Π1.</p>},
	url = {https://arxiv.org/abs/2110.04651},
	author = {Hamoon Mousavi and Seyed Sajjad Nezhadi and Henry Yuen}
}
@article {1934,
	title = {Raz-McKenzie simulation with the inner product gadget},
	journal = {Electronic Colloquium on Computational Complexity (ECCC)},
	year = {2017},
	month = {2017/01/28},
	abstract = {<p>In this note we show that the Raz-McKenzie simulation algorithm which lifts deterministic query lower bounds to deterministic communication lower bounds can be implemented for functions f composed with the Inner Product gadget 1ip(x, y) = P i xiyi mod 2 of logarithmic size. In other words, given a function f : {0, 1} n \&rarr; {0, 1} with deterministic query complexity D(f), we show that the deterministic communication complexity of the composed function f {\textopenbullet} 1 n ip is Θ(D(f) log n), where f {\textopenbullet} 1 n ip(x, y) = f(1ip(x 1 , y 1 ), . . . , 1ip(x n , y n )) where x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) and each x i and y i are O(log n) bit strings. In [RM97] and [GPW15], the simulation algorithm is implemented for functions composed with the Indexing gadget, where the size of the gadget is polynomial in the input length of the outer function f.</p>},
	url = {https://eccc.weizmann.ac.il/report/2017/010/},
	author = {Xiaodi Wu and Penghui Yao and Henry Yuen}
}
