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arXiv:2608.02594

An Argmax Principle for Sum-of-Squares Relaxations on the Sphere

Fernando Jeronimo Granha, Pei Wu, Haochen Xu

cs.CCquant-phSubmitted Aug 3, 2026• Announced Aug 4, 2026• 0 citations
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Abstract

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 $Φ_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}$. 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-$O(\sqrt{n/ε})$ SoS analysis for approximating $h_{\mathrm{sep}}(P)$ 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 $\mathrm{QMA}(2)$ protocols. Second, for the matrix $2\to4$ norm, degree-$O(\sqrt n/ε)$ 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 $p\to q$ 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 $O_d((n/k)^{d/2-1})$. 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.

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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⟩2kΦ_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}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)O(\sqrt{n/ε})O(n/ε​) SoS analysis for approximating hsep(P)h_{\mathrm{sep}}(P)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)\mathrm{QMA}(2)QMA(2) protocols. Second, for the matrix 2→42\to42→4 norm, degree-O(n/ε)O(\sqrt n/ε)O(n​/ε) SoS gives a multiplicative (1+ε)(1+ε)(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→qp\to qp→q norms with even qqq. Finally, for degree-ddd polynomial optimization, we recover the convergence theorem of Bhattiprolu et al. (FOCS'17) with a shorter, more direct proof: degree-kkk SoS gives approximation ratio Od((n/k)d/2−1)O_d((n/k)^{d/2-1})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.
We prove optimal finite quantum de Finetti upper bounds. Given a bosonic state ρN∈D(SymN(Cd))ρ_N\in D(\mathrm{Sym}^N(\mathbb C^d))ρN​∈D(SymN(Cd)), there is a probability measure ννν on the unit sphere such that ∥ρN(2)−∫∣u⟩⟨u∣⊗2 dν(u)∥1≤d−1N−1. \left\| ρ_N^{(2)}-\int |u\rangle\langle u|^{\otimes 2}\,dν(u) \right\|_1 \le \frac{\sqrt{d-1}}{N-1}. ​ρN(2)​−∫∣u⟩⟨u∣⊗2dν(u)​1​≤N−1d−1​​. By purification, the bosonic theorem also gives the optimal O(d/N)O(d/N)O(d/N) upper bound for arbitrary exchangeable states. These results settle the dimension dependence left open by Christandl, König, Mitchison, and Renner (CMP 2007). The proof casts de Finetti approximation as sum-of-squares rounding and applies the argmax method of Jeronimo, Wu, and Xu (manuscript 2026). More generally, ttt-site marginals satisfy O(td/N)O(t\sqrt d/N)O(td​/N) bosonic and O(td/N)O(td/N)O(td/N) permutation-invariant bounds. Our proof formulates de Finetti approximation as the integrality gap of a symmetric-extension semidefinite program and rounds an optimum by the argmax principle. The sharp bounds have several consequences. For every fixed ε∈(0,1)\varepsilon\in(0,1)ε∈(0,1), we construct a channel with input dimension D=exp⁡(Oε(dlog⁡d))=exp⁡(o(d))D=\exp(O_\varepsilon(\sqrt d\log d))=\exp(o(d))D=exp(Oε​(d​logd))=exp(o(d)) whose outputs are ε\varepsilonε-close to separable states of local dimension ddd and whose image contains every such separable state, thereby refuting Watrous's disentangler conjecture. We also obtain deterministic exp⁡(O~(d/ε))\exp(\widetilde O(\sqrt d/\varepsilon))exp(O(d​/ε))-time algorithms for explicit Best Separable State without perfect completeness and for trace-distance separability testing. Finally, spectral truncation gives the first dimension-free bosonic de Finetti theorem in Hilbert--Schmidt distance, with the optimal rate Θ(N−1/2)Θ(N^{-1/2})Θ(N−1/2) when the dimension may grow.
We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof system to transform a *combining algorithm* -- an algorithm that maps a distribution over solutions into a (possibly weaker) solution -- into a *rounding algorithm* that maps a solution of the relaxation to a solution of the original problem. Using this approach, we obtain algorithms that yield improved results for natural variants of three well-known problems: 1) We give a quasipolynomial-time algorithm that approximates the maximum of a low degree multivariate polynomial with non-negative coefficients over the Euclidean unit sphere. Beyond being of interest in its own right, this is related to an open question in quantum information theory, and our techniques have already led to improved results in this area (Brandão and Harrow, STOC '13). 2) We give a polynomial-time algorithm that, given a d dimensional subspace of R^n that (almost) contains the characteristic function of a set of size n/k, finds a vector vvv in the subspace satisfying ∣v∣44>c(k/d1/3)∣v∣22|v|_4^4 > c(k/d^{1/3}) |v|_2^2∣v∣44​>c(k/d1/3)∣v∣22​, where ∣v∣p=(Eivip)1/p|v|_p = (E_i v_i^p)^{1/p}∣v∣p​=(Ei​vip​)1/p. Aside from being a natural relaxation, this is also motivated by a connection to the Small Set Expansion problem shown by Barak et al. (STOC 2012) and our results yield a certain improvement for that problem. 3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time algorithm with substantially improved guarantees for recovering a planted μμμ-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n nonzero coordinates, we can recover it with high probability whenever μ<O(min⁡(1,n/d2))μ< O(\min(1,n/d^2))μ<O(min(1,n/d2)), improving for d<n2/3d < n^{2/3}d<n2/3 prior methods which intrinsically required μ<O(1/(d))μ< O(1/\sqrt(d))μ<O(1/(​d)).
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).
Chvátal--Gomory cuts (CG-cuts) and the Bienstock--Zuckerberg hierarchy capture useful linear programs that the standard bounded degree sum-of-squares (SoS) hierarchy fails to capture. In this paper we present a novel polynomial time SoS hierarchy for 0/1 problems with a custom subspace of high degree polynomials (not the standard subspace of low degree polynomials). We show that the new SoS hierarchy recovers the Bienstock--Zuckerberg hierarchy. Our result implies a linear program that reproduces the Bienstock--Zuckerberg hierarchy as a polynomial-sized, efficiently constructible extended formulation that satisfies all constant pitch inequalities. The construction is also very simple, and it is fully defined by giving the supporting polynomials. Moreover, for a class of polytopes (e.g., set cover and packing problems), the resulting SoS hierarchy optimizes in polynomial time over the polytope resulting from any constant rounds of CG-cuts, up to an arbitrarily small error in the solution value. Arguably, this is the first example where different basis functions can be useful in asymmetric situations to obtain a hierarchy of relaxations.
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/\ell ^2)O(d2/ℓ2) O ( d 2 / ℓ 2 ) in the regime ℓ=Ω(d)\ell = \Omega (d)ℓ=Ω(d) ℓ = Ω ( d ) where ℓ\ell 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.
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.