12,652 research outputs found
On the Bit Complexity of Sum-of-Squares Proofs
It has often been claimed in recent papers that one can find a degree d Sum-of-Squares proof if one exists via the Ellipsoid algorithm. In a recent paper, Ryan O\u27Donnell notes this widely quoted claim is not necessarily true. He presents an example of a polynomial system with bounded coefficients that admits low-degree proofs of non-negativity, but these proofs necessarily involve numbers with an exponential number of bits, causing the Ellipsoid algorithm to take exponential time. In this paper we obtain both positive and negative results on the bit complexity of SoS proofs.
First, we propose a sufficient condition on a polynomial system that implies a bound on the coefficients in an SoS proof. We demonstrate that this sufficient condition is applicable for common use-cases of the SoS algorithm, such as Max-CSP, Balanced Separator, Max-Clique, Max-Bisection, and Unit-Vector constraints.
On the negative side, O\u27Donnell asked whether every polynomial system containing Boolean constraints admits proofs of polynomial bit complexity. We answer this question in the negative, giving a counterexample system and non-negative polynomial which has degree two SoS proofs, but no SoS proof with small coefficients until degree sqrt(n)
A generalization of CHSH and the algebraic structure of optimal strategies
Self-testing has been a rich area of study in quantum information theory. It
allows an experimenter to interact classically with a black box quantum system
and to test that a specific entangled state was present and a specific set of
measurements were performed. Recently, self-testing has been central to
high-profile results in complexity theory as seen in the work on entangled
games PCP of Natarajan and Vidick (FOCS 2018), iterated compression by
Fitzsimons et al. (STOC 2019), and NEEXP in MIP* due to Natarajan and Wright
(FOCS 2019).
In this work, we introduce an algebraic generalization of CHSH by viewing it
as a linear constraint system (LCS) game, exhibiting self-testing properties
that are qualitatively different. These provide the first example of non-local
games that self-test non-Pauli operators resolving an open questions posed by
Coladangelo and Stark (QIP 2017). Our games also provide a self-test for states
other than the maximally entangled state, and hence resolves the open question
posed by Cleve and Mittal (ICALP 2012). Additionally, our games have bit
question and bit answer lengths making them suitable candidates for
complexity theoretic application. This work is the first step towards a general
theory of self-testing arbitrary groups. In order to obtain our results, we
exploit connections between sum of squares proofs, non-commutative ring theory,
and the Gowers-Hatami theorem from approximate representation theory. A crucial
part of our analysis is to introduce a sum of squares framework that
generalizes the \emph{solution group} of Cleve, Liu, and Slofstra (Journal of
Mathematical Physics 2017) to the non-pseudo-telepathic regime. Finally, we
give the first example of a game that is not a self-test. Our results suggest a
richer landscape of self-testing phenomena than previously considered.Comment: Incorporated reviewers comments and fixed typo
Quantum de Finetti Theorems under Local Measurements with Applications
Quantum de Finetti theorems are a useful tool in the study of correlations in
quantum multipartite states. In this paper we prove two new quantum de Finetti
theorems, both showing that under tests formed by local measurements one can
get a much improved error dependence on the dimension of the subsystems. We
also obtain similar results for non-signaling probability distributions. We
give the following applications of the results:
We prove the optimality of the Chen-Drucker protocol for 3-SAT, under the
exponential time hypothesis.
We show that the maximum winning probability of free games can be estimated
in polynomial time by linear programming. We also show that 3-SAT with m
variables can be reduced to obtaining a constant error approximation of the
maximum winning probability under entangled strategies of O(m^{1/2})-player
one-round non-local games, in which the players communicate O(m^{1/2}) bits all
together.
We show that the optimization of certain polynomials over the hypersphere can
be performed in quasipolynomial time in the number of variables n by
considering O(log(n)) rounds of the Sum-of-Squares (Parrilo/Lasserre) hierarchy
of semidefinite programs. As an application to entanglement theory, we find a
quasipolynomial-time algorithm for deciding multipartite separability.
We consider a result due to Aaronson -- showing that given an unknown n qubit
state one can perform tomography that works well for most observables by
measuring only O(n) independent and identically distributed (i.i.d.) copies of
the state -- and relax the assumption of having i.i.d copies of the state to
merely the ability to select subsystems at random from a quantum multipartite
state.
The proofs of the new quantum de Finetti theorems are based on information
theory, in particular on the chain rule of mutual information.Comment: 39 pages, no figure. v2: changes to references and other minor
improvements. v3: added some explanations, mostly about Theorem 1 and
Conjecture 5. STOC version. v4, v5. small improvements and fixe
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