3,658 research outputs found

    Limitations of semidefinite programs for separable states and entangled games

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    Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no ω(1)\omega(1)-round integrality gaps were known: the set of separable (i.e. unentangled) states, or equivalently, the 2→42 \rightarrow 4 norm of a matrix, and the set of quantum correlations; i.e. conditional probability distributions achievable with local measurements on a shared entangled state. In both cases no-go theorems were previously known based on computational assumptions such as the Exponential Time Hypothesis (ETH) which asserts that 3-SAT requires exponential time to solve. Our unconditional results achieve the same parameters as all of these previous results (for separable states) or as some of the previous results (for quantum correlations). In some cases we can make use of the framework of Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not only the SoS hierarchy. Our hardness result on separable states also yields a dimension lower bound of approximate disentanglers, answering a question of Watrous and Aaronson et al. These results can be viewed as limitations on the monogamy principle, the PPT test, the ability of Tsirelson-type bounds to restrict quantum correlations, as well as the SDP hierarchies of Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published versio

    Tight Sum-of-Squares lower bounds for binary polynomial optimization problems

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    We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree 2d2d and an odd number of variables nn, we prove that n+2d−12\frac{n+2d-1}{2} levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires nn levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is n−1n-1. We disprove this conjecture and derive lower and upper bounds for the rank

    Equivariant semidefinite lifts and sum-of-squares hierarchies

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    A central question in optimization is to maximize (or minimize) a linear function over a given polytope P. To solve such a problem in practice one needs a concise description of the polytope P. In this paper we are interested in representations of P using the positive semidefinite cone: a positive semidefinite lift (psd lift) of a polytope P is a representation of P as the projection of an affine slice of the positive semidefinite cone S+d\mathbf{S}^d_+. Such a representation allows linear optimization problems over P to be written as semidefinite programs of size d. Such representations can be beneficial in practice when d is much smaller than the number of facets of the polytope P. In this paper we are concerned with so-called equivariant psd lifts (also known as symmetric psd lifts) which respect the symmetries of the polytope P. We present a representation-theoretic framework to study equivariant psd lifts of a certain class of symmetric polytopes known as orbitopes. Our main result is a structure theorem where we show that any equivariant psd lift of size d of an orbitope is of sum-of-squares type where the functions in the sum-of-squares decomposition come from an invariant subspace of dimension smaller than d^3. We use this framework to study two well-known families of polytopes, namely the parity polytope and the cut polytope, and we prove exponential lower bounds for equivariant psd lifts of these polytopes.Comment: v2: 30 pages, Minor changes in presentation; v3: 29 pages, New structure theorem for general orbitopes + changes in presentatio

    Lower bounds on the size of semidefinite programming relaxations

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    We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on nn-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nc2^{n^c}, for some constant c>0c > 0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1)O(1) sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for MAX-3-SAT

    Equivariant semidefinite lifts of regular polygons

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    Given a polytope P in Rn\mathbb{R}^n, we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the linear projection of an affine slice of the positive semidefinite cone S+d\mathbf{S}^d_+. If a polytope P has symmetry, we can consider equivariant psd lifts, i.e. those psd lifts that respect the symmetry of P. One of the simplest families of polytopes with interesting symmetries are regular polygons in the plane, which have played an important role in the study of linear programming lifts (or extended formulations). In this paper we study equivariant psd lifts of regular polygons. We first show that the standard Lasserre/sum-of-squares hierarchy for the regular N-gon requires exactly ceil(N/4) iterations and thus yields an equivariant psd lift of size linear in N. In contrast we show that one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1, which is exponentially smaller than the psd lift of the sum-of-squares hierarchy. Our construction relies on finding a sparse sum-of-squares certificate for the facet-defining inequalities of the regular 2^n-gon, i.e., one that only uses a small (logarithmic) number of monomials. Since any equivariant LP lift of the regular 2^n-gon must have size 2^n, this gives the first example of a polytope with an exponential gap between sizes of equivariant LP lifts and equivariant psd lifts. Finally we prove that our construction is essentially optimal by showing that any equivariant psd lift of the regular N-gon must have size at least logarithmic in N.Comment: 29 page
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