Abstract We consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of sum-of-squares relaxations. Exploiting the polynomial kernel technique, we obtain a quadratic improvement of the known convergence rate by Reznick and Doherty and Wehner. Specifically, we show that the rate of convergence is no worse than $$O(d^2/\ell ^2)$$ O ( d 2 / ℓ 2 ) in the regime $$\ell = \Omega (d)$$ ℓ = Ω ( d ) where $$\ell $$ ℓ is the level of the hierarchy and d the dimension, solving a problem left open in the recent paper by de Klerk and Laurent ( arXiv:1904.08828 ). Importantly, our analysis also works for matrix-valued polynomials on the sphere which has applications in quantum information for the Best Separable State problem. By exploiting the duality relation between sums of squares and the Doherty–Parrilo–Spedalieri hierarchy in quantum information theory, we show that our result generalizes to nonquadratic polynomials the convergence rates of Navascués, Owari and Plenio.
Abstract We consider the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of sum-of-squares relaxations. Exploiting the polynomial kernel technique, we obtain a quadratic improvement of the known convergence rate by Reznick and Doherty and Wehner. Specifically, we show that the rate of convergence is no worse than O(d2/ℓ2) O ( d 2 / ℓ 2 ) in the regime ℓ=Ω(d) ℓ = Ω ( d ) where ℓ ℓ is the level of the hierarchy and d the dimension, solving a problem left open in the recent paper by de Klerk and Laurent ( arXiv:1904.08828 ). Importantly, our analysis also works for matrix-valued polynomials on the sphere which has applications in quantum information for the Best Separable State problem. By exploiting the duality relation between sums of squares and the Doherty–Parrilo–Spedalieri hierarchy in quantum information theory, we show that our result generalizes to nonquadratic polynomials the convergence rates of Navascués, Owari and Plenio.
Computing the optimal success probability for transmitting classical messages through a single use of a quantum channel is NP-hard, even for two messages. An existing semidefinite programming hierarchy based on symmetric extensions provides convergent upper bounds with an a priori error estimate that decays as the inverse square root of the extension level. In this work, we construct a Hermitian sum-of-squares hierarchy for an arbitrary number of messages and prove quadratic convergence in its level. The error bound is proportional to the advantage over random guessing. Our approach combines state-discrimination duality with positive polynomial kernels on products of spheres to construct feasible polynomial dual certificates. For binary messages, the resulting bounds give a multiplicative approximation from above of the trace-norm contraction coefficient.
Abstract. We revisit the convergence analysis of two approximation hierarchies for polynomial optimization on the unit sphere. The first one is based on the moment-sos approach and gives semidefinite bounds for which Fang and Fawzi (2021) showed an analysis in [Formula: see text] for the [Formula: see text]th level bound, using the polynomial kernel method. The second hierarchy was recently proposed by Lovitz and Johnston (2023) and gives spectral bounds for which they show a convergence rate in [Formula: see text], using a quantum de Finetti theorem of Christandl et al. (2007) that applies to complex Hermitian matrices with a “double” symmetry. We investigate links between these approaches, in particular, via duality of moments and sums of squares. Our main results include showing that the spectral bounds cannot have a convergence rate better than [Formula: see text] and that they do not enjoy generic finite convergence. In addition, we propose alternative performance analyses that involve explicit constants depending on intrinsic parameters of the optimization problem. For this we develop a novel “banded” real de Finetti theorem that applies to real matrices with “double” symmetry. We also show how to use the polynomial kernel method to obtain a de Finetti type result in [Formula: see text] for real maximally symmetric matrices, improving an earlier result in [Formula: see text] of Doherty and Wehner (2012).
Moment/Sum-of-Hermitian-Squares relaxations for noncommutative polynomial optimization problems have become an important tool for analyzing problems within quantum theory. Despite their widespread success, little is known about their rate of convergence and, consequently, their accuracy. In this work, we develop explicit convergence rates for relaxations of noncommutative polynomial optimization problems generated from the Pauli algebra -- covering applications to the ground state energy problem for n-qubit systems. In particular, we show that the rate of convergence can be bounded in terms of the smallest roots of a family of orthogonal polynomials known as Krawtchouk polynomials. Our result represents the first quantitative analysis of the rate of convergence for relaxations of noncommutative polynomial optimization problems.
We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere. Given a feasible pseudo-expectation, we form a polynomial of high-order pseudo-moments, such as Φk(u)=E⟨x,u⟩2k. Our guiding principle is that its maximizers are rounding candidates: their local and global optimality conditions reveal the reweighed pseudo-expectation inequalities governing SoS convergence. This viewpoint unifies several problems previously analyzed by rather different techniques.
We obtain three results. First, for Best Separable State, we give a degree-O(n/ε) SoS analysis for approximating hsep(P) in the perfect-completeness regime, improving and simplifying Barak, Kothari and Steurer (STOC'17). The dependence is essentially tight for inverse-linear gap under the Exponential-Time Hypothesis, matching hardness from QMA(2) protocols. Second, for the matrix 2→4 norm, degree-O(n/ε) SoS gives a multiplicative (1+ε) approximation. Barak et al. (STOC'12) previously gave a comparable-time constant-gap decision algorithm; our result gives a multiplicative guarantee and extends to a family of p→q norms with even q. Finally, for degree-d polynomial optimization, we recover the convergence theorem of Bhattiprolu et al. (FOCS'17) with a shorter, more direct proof: degree-k SoS gives approximation ratio Od((n/k)d/2−1).
The paper introduces no new relaxation. Instead, the high-moment argmax gives a common way to read an SoS solution, unifying previously separate convergence analyses and yielding sharper bounds or simpler proofs.
In this paper, we study the discrete complex quadratic optimization problem, where the goal is to maximize a multivariate Hermitian quadratic objective function over complex m’th roots of unity. This problem is a generalization of the well-studied quadratic unconstrained binary optimization problem (QUBO), and has applications in signal processing, including in maximal likelihood estimation and multiple-input multiple-output detection, in addition to applications in statistical physics, in particular the computation of ground states of the Potts model. We study the sum-of-squares hierarchy for this problem, which is a sequence of semidefinite relaxations that is known to converge to the optimal solution after at most n steps. A previous result of Fawzi, Saunderson and Parrilo showed that in the the binary case ( m=2 ), at most ⌈n/2⌉ levels of the hierarchy are actually needed. In this paper we generalize this result and establish that the hierarchy converges after at most ⌊n/2⌋+1 level for any m≥2 . Our proof generalizes the ideas used in the m=2 case, and is based on constructing chordal covers of appropriate Cayley graphs.
The noncommutative sum-of-squares (ncSoS) hierarchy was introduced by Navascués-Pironio-Acín as a sequence of semidefinite programming relaxations for approximating values of noncommutative polynomial optimization problems, which were originally intended to generalize quantum values of nonlocal games. Recent work has started to analyze the hierarchy for approximating ground energies of local Hamiltonians, initially through rounding algorithms which output product states for degree-2 ncSoS applied to Quantum Max-Cut. Some rounding methods are known which output entangled states, but they use degree-4 ncSoS. Based on this, Hwang-Neeman-Parekh-Thompson-Wright conjectured that degree-2 ncSoS cannot beat product state approximations for Quantum Max-Cut and gave a partial proof relying on a conjectural generalization of Borrell's inequality. In this work we consider a family of Hamiltonians (called the quantum rotor model in condensed matter literature or lattice O(k) vector model in quantum field theory) with infinite-dimensional local Hilbert space L2(Sk−1), and show that a degree-2 ncSoS relaxation approximates the ground state energy better than any product state.